Design Column Using Eurocode 2 Worked Examples
The present paper proposes a simplified method to design slender rectangular reinforced concrete columns with doubly symmetric reinforcement. The proposal is based on the computation of the second-order eccentricity method from the Eurocode 2 ( December 2004). It is valid for columns subjected to combined axial loads and either uniaxial or biaxial bending, short-time and sustained loads, and also for normal- and high-strength concretes. It is only suitable for columns with equal effective buckling lengths in the two principal bending planes. It is an extension for biaxial bending of the column-model method. The current paper is the second part of a research study conducted by the current authors. The method was compared with 371 experimental tests from the literature and a high degree of accuracy was obtained. Precision for sustained loads and biaxial bending was improved in comparison with the method proposed by Eurocode 2 ( December 2004). The method allows slender reinforced concrete columns to be both checked and designed with sufficient accuracy for engineering practice.
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Design method for slender columns subjected to
biaxial bending based on second-order
eccentricity
J. L. Bonet,* M. L. Romero,* M. A. Fernandez* and P. F. Miguel*
Technical University of Valencia
The present paper proposes a simplified method to design slender rectangular reinforced concrete columns with
doubly symmetric reinforcement. The proposal is based on the computation of the secon d-order eccentricity method
from the Eurocode 2 (December 2004). It is valid for columns subjected to combined axial loads and either uniaxial
or biaxial bending, short-time and sustained loads, and also for normal- and high-strength concretes. It is only
suitable for columns with equal effective buckling lengths in the two principal bending planes. It is an extension for
biaxial bending of the column-model method. The current paper is the second part of a research study conducted
by the current authors. The method was compared with 371 experimental tests from the literature and a high degree
of accuracy was obtained. Precision for sustained loads and biaxial bending was im proved in comparison with the
method proposed by Eurocode 2 (December 2004). The method allows slender reinforced concrete columns to be
both checked and designed with sufficient accuracy for engineering practice.
Notation
b, h width and depth of the rectangular
section
d
eq
equivalent effective depth
d
y
, d
z
effective depth with respect to y- and
z-axis respectively: d
z
¼ h/2 + i
sz
;d
y
¼
b/2 + i
s y
E
s
elastic modulus of the longitudinal
reinforcement
e
Ed
¼ e
0Ed
+ e
2
; vector modulus of the total
design eccentricity
e
0Ed
vector modulus of the first-order
eccentricity
e
0Ed y
, e
0Ed z
first-order eccentricity about y- and
z-axis respectively
e
2
second-order eccentricity
h
c
critical dimension of the cross-section
i
s y
, i
s z
radii of gyration of the reinforcement
with respect to y- and z-axis
respectively
K
j
correction factor of the curvature for
taking account of the long-term effects
K
c
correction factor of the curvature
l
0
effective length of the column
M
0Ed
vector modulus of the first-order
bending moment of the column
M
0Ed y
, M
0Ed z
first-order bending moments of the
column in the direction y and z
respectively
M
E d
vector modulus of the design bending
moment
M
Ed y
, M
Ed z
design moment about y and z axes
respectively
M
0Eqp
vector modulus of the first-order
bending moment in the quasi-permanent
load combination
N
Ed
design value of the axial load
biaxial bending angle with respect to
the strong axis
* relative biaxial bending angle with
respect to the strong axis
interpolation function to obtain the
equivalent effective depth (d
eq
)
cu2
¼ (2
.
6 + 35 [(90 f
ck
)/100]
4
)/1000
< 0
.
0035 ultimate strain of the concrete
for bending and axial load
* Campus de Vera s/n. Technical University of Valencia. 46022
Valencia. Spain.
(MCR 51410) Paper received 12 May 2005; revised 12 January 2006;
accepted for publication 12 May 2006.
Magazine of Concrete Research, 2007, 59, No. 1, February, 3–19
3
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yd
¼ f
yd
/E
s
strain correspondent with the
yielding stress of steel
º
g
¼ l
0
/h
c
, geometric slenderness ratio of
the column
j creep coefficient
j
ef
j(M
0Eqp
/M
0Ed
) effective creep ratio
1/r nominal curvature
1/r
0
base curvature
Introduction
The utilisation of high-strength concrete for civil and
building structures has become more common in recent
years. The use of such material allows the size of the
sections to be reduced while maintaining the same
strength capacity in comparison with normal-strength
columns (less than 50 MPa). This reduction produces
an increase in the slenderness that has to be considered
properly in the analysis.
The design of slender reinforced concrete columns is
difficult because the nonlinear behaviour of the materi-
als and the equilibrium of the structure in the deformed
shape (nonlinear geometry) must both be taken into
account.
General nonlinear methods of analysis with numeri-
cal approximations, as used by Mari,
1
Wang and Hsu
2
and Ahmad and Weerakoon
3
are of no use for everyday
design because they require previous knowledge of cer-
tain data, which are initially unknown (such as the area
of the reinforcing bars), and also they are computation-
ally intensive since they require solving many coupled
nonlinear equations many times.
4
A number of authors
are therefore interested in simplified methods.
5,6
Most design codes suggest the utilisation of simpli-
fied methods that are helpful in the design process of
columns under uniaxial bending. However, the current
methods in the codes were developed for normal-
strength concretes. Generally, most European codes,
such as BAEL-91,
7
BS 8110,
8
EC-2,
9
, MC-90,
10
and
EHE,
11
design the cross-section for a total eccentricity
(e
Ed
), obtained as the addition of the first-order eccen-
tricity (e
0Ed
) and the second-order eccentricity (e
2
),
which takes into account the second-order effects. The
first-order eccentricity is equal to the ratio between the
first-order bending moment (M
0Ed
) and the design value
of the axial load (N
Ed
).
The second-order eccentricity is proportional to the
nominal curvature (1/r) and the square of the effective
buckling length (l
0
) of the column. The nominal curva-
ture (1/r) depends on different factors such as the
cracking, the creep and the nonlinear behaviour of the
materials. Over the past 40 years, numerous proposals
have been put forward by different authors such as van
Laruwen and van Riel,
12
Cranston,
13
Beal and Khalil,
5
and Westerberg.
14
Most of them propose to calculate
the nominal curvature (1/r) as the product of a base
curvature (1/r
o
) and a correction factor (K
c
), which
depends on the forces on the column and the long-term
effects. For the draft of the Eurocode EC-2
15
and the
MC-90
10
and for sections with symmetric reinforce-
ment concentrated at the top and the bottom, the base
curvature denotes the initial yielding state of the col-
umn critical section. It is equivalent to the state of
strains that produce the simultaneous yielding of the
compression and tension reinforcement bars of the sec-
tion. Nevertheless, for the French code BAEL-91,
7
Cranston
13
and the CEB,
16
the base curvature denotes,
for the same type of sections, the strain state where
simultaneous yielding of the most highly tensioned
reinforced bar is produced and the concrete reaches the
ultimate strain (
cu2
, for bending and axial load). Fol-
lowing this procedure, the current paper proposes a
technique to compute the second-order eccentricity (e
2
)
that is valid for both normal- and high-strength con-
cretes.
Moreover, many reinforced concrete columns are
subjected to biaxial bending and axial loads as a result
of their position in the structure, the shape of the cross-
section or the source of the external loads. For those
cases, and for rectangular, circular or elliptical col-
umns, the draft of the EC-2
15
computes the second-
order eccentricity separately in each direction of the
principal axes and the design is performed using the
'load contour method' by Bresler.
17
M
Ed y
M
Rd y
a
þ
M
Ed z
M
Rd z
a
< 1 (1)
where M
Rd y
, M
Rd z
are the moment resistance in the
direction y and z axes, respectively; M
Ed y
, M
Ed z
are the
design moments that are applied in the critical cross-
section of the support, including a nominal second-
order moment; and a is the axial load contour expo-
nent. For circular or elliptical sections a ¼ 2, and for
rectangular sections:
(a) N
Ed
/N
Rd
: <0
.
1, 0
.
7, 1
.
0
(b) a ¼ 1
.
0, 1
.
5, 2
.
0
N
Ed
is the design value of the axial load; N
Rd
¼ 0
.
85
.
f
cd
.
A
c
+A
s
.
f
yd
, design axial resistance of section; A
c
,
A
s
are the gross area of the concrete sections and the
longitudinal reinforcement; and f
cd
,f
yd
represents the
design strength of concrete and steel.
According to Bonet et al.,
18
this method can give
rise to unsafe situations for axial load levels close to
the ultimate axial load of the column if the most
important bending force corresponds to the direction of
the lower slenderness (bending with respect to the
strong axis). This problem is owing to the fact that the
'load contour method' does not take into account the
interaction that both curvatures produce in the structur-
al behaviour of the support.
Further, it is important to emphasise the fact that the
method from EC-2
15
needs previous knowledge of the
amount of reinforcement of the column, so that an
Bonet et al.
4
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iterative process is required when the method is applied
to design the reinforcing bars.
The method proposed in the current paper includes
the interaction that exists between both flexural axes in
the structural behaviour of the column, and it is applic-
able for both normal- and high-strength concretes. In
addition it is a direct method because it does not
depend on the value of the mechanical reinforcement
ratio.
The columns studied here are isolated elements with
pinned ends subjected to constant axial load and biaxial
bending valid both for short-term and sustained loads,
(Fig. 1). Other effects such as different end restraints,
loading conditions and lateral supports are accounted
for in the draft of the EC-2
15
through the use of the
effective length factor (K) and the equivalent first-order
end moment (M
0e
).
Objectives
The present paper has two objectives. The first is to
propose a new equation to calculate the second-order
eccentricity (e
2
) of slender reinforced concrete columns
for single curvature. The second is to put forward a
new simplified method, termed the 'second-order biax-
ial eccentricity method', for designing slender columns
with equal effective lengths in both directions that are
subjected to axial loads and biaxial bending, and which
is based on the calculus of the second-order eccentri-
city taking into account the interaction between both
bending axes. This is an extension for biaxial bending
of the column-model method.
The proposed equation of the second-order eccentri-
city e
2
is applicable to a high percentage of rectangular
sections, with both short-term and sustained loads, and
for normal- and high-strength concretes. The current
paper is the second part of a research study conducted
by the present authors, Bonet et al.
19
Method
The proposed method is based on the calculation of
the total design eccentricity (e
Ed
) obtained from the
addition of the vector modulus of the first-order eccen-
tricity (e
0 Ed
) and the second-order eccentricity (e
2
)
e
Ed
¼ e
0Ed
þ e
2
(2)
where
e
0Ed
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
e
2
0Ed y
þ e
2
0Ed z
q
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
M
0Ed z
= N
Ed
ðÞ
2
þ M
0Ed y
= N
Ed
2
q
(3)
M
0Ed y
,M
0Ed z
are the first-order bending moments of
the column (Fig. 1)
e
2
is the second-order eccentricity
e
2
¼
1
r
l
2
0
c
(4)
where c ¼ 10 for sinusoidal curvature distribution, as it
is stated in the model-column method, and 1/r is nom-
inal curvature
In the sections that follow, the equation of (1/ r) will
be obtained from a numerical simulation and will later
M
0Edy
5 N
Ed
· e
0Edz
z
y
M
Ed
M
Edz
b
h
y
l
0
N
Ed
N
Ed
z
z
e
0Edz
e
0Edz
e
0Edy
Weak
axis
Strong
axis
y
e
0Edy
M
0Edz
5 N
Ed
· e
0Edy
â
M
0Edz
M
0Ed
M
0Edy
M
Edy
Fig.1. The proposed simplified method
Design method for slender columns
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be compared with experimental tests from the litera-
ture.
The cross-section is designed for a factored axial
load (N
Ed
) and a total design bending moment (M
Ed
)
obtained as a product between N
Ed
and the total eccen-
tricity (e
Ed
). This bending moment will have the same
bending direction as the first-order bending moment
applied (Fig. 1).
M
Ed
¼ N
Ed
e
Ed
(5)
Numerical simulation
The equation of the nominal curvature (1/r )was
inferred from using a general method of structural
analysis for reinforced concrete using finite elements.
This numerical method includes the following main
issues.
(a) one-dimensional finite element with non-constant
curvature
1
(b) nonlinear concrete behaviour
10,20
(c) nonlinear steel behaviour: bilinear diagram
10
(d) geometric nonlinearity: large displacements and
large deformations
(e) time-dependent effects: creep and shrinkage.
16,21
A more thorough description of the model can be found
in Bonet et al.
18
The foregoing numerical model was used here to
perform the analysis of the main variables that exert an
influence on nominal curvature (1/r).
Table 1 shows the parameters that were analysed and
their variation coefficients, which when combined pro-
duced 7600 numerical tests. The mechanical reinforce-
ment cover was fixed at 10% of the height and the
width of the section. This table is similar to that in
Bonet et al.
19
but, in this case, the nominal curvature
(1/r) is the objective of the research.
For the particular case of rectangular sections with
reinforcement equal at the four faces, only one octant
(458) of the interaction surface has to be studied. For
this case, the following angles were selected ( ): 08,
158 ,30 8 and 458. On the other hand, for a general
rectangular section, 908 of the interaction surface need
to be studied. For this case, the following angles were
selected: the boundary angles 08 and 908, the angle
corresponding to the load where relative bending mo-
ments are equal (M
0Ed y
/h ¼ M
0Ed z
/b), and two inter-
mediate angle values.
Proposal of nominal curvature 1/r
Nominal curvature (1/r) of a column for axial loads
and uniaxial bending under short-term loads
The estimation of nominal curvature is obtained
through the following equation
1
r
¼ K
c
1
r
0
(6)
where K
c
is a correction factor of the curvature,
1/r
0
represents base curvature (Fig. 2)
1
r
0
¼
cu2
þ
yd
h=2 þ i
s
(7)
yd
is strain correspondent with the yielding stress of
steel ( f
yd
)
yd
¼
f
yd
E
s
(8)
E
s
is the elastic modulus of the longitudinal reinforce-
ment, h is the height of the section following the bend-
ing direction of the column, i
s
is the radius of gyration
Table 1. Parameter variation
Parametric Values
Column geometric slenderness (º
g
) º
g
¼ 10, 15, 20, 25, 30
Cross-section shape Rectangular
Height–width ratio (h/b) h/b ¼ 1, 1
.
5 and 2
Biaxial bending angle ( ) with respect to the strong axis (Fig. 1) For h/b ¼ 1, ¼ 08 ,15 8 ,30 8 and 458
For h/b ¼ 1
.
5, ¼ 08 ,17 8,34 8,62 8 and 908
For h/b ¼ 2, ¼ 08 ,14 8,27 8,59 8 and 908
Reinforcement distribution Doubly symmetric at four corners
Doubly symmetric and uniformly distributed at four faces
Symmetric at opposite faces
Structural typology Isolated element with pinned ends
Axial load Ten values for equivalent steps, starting from a zero axial load to the
ultimate capacity for pure compression
Compressive concrete strength ( f
c
) f
c
¼ 30 MPa, 50 MPa and 80 MPa
Steel strength ( f
y
) f
y
¼ 400 MPa and 500 MPa
Mechanical reinforcement ratio (ø ) ø ¼ 0
.
06, 0
.
25, 0
.
50, 0
.
75
Creep coefficient (j) j ¼ 1, 2, 3
Bonet et al.
6
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of the reinforcements with respect to the centroid of
the concrete cross-section (see Appendix 1),
cu2
is the
ultimate strain of the concrete for bending and axial
load, Art 3.1.7. EC-2
15
cu2
(‰) ¼ 2
:
6 þ 35 90 f
ck
ðÞ
=10
4
< 3
:
5 (9)
or from Table 2, and f
ck
is the characteristic compres-
sive strength of concrete.
The base curvature (1/r
o
) selected for the particular
case where reinforcement is concentrated at the oppo-
site faces of the section (Fig. 2) corresponds to the
critical state by which the longitudinal reinforcement
bar under tension yields (
yd
) and the concrete reaches
the ultimate strain (
cu2
).
It should be pointed out that the ultimate strain (
cu2
)
was selected from the proposal of the draft of EC-2
15
for any of the design stress–strain diagrams (parabola–
rectangle, bilinear or equivalent stress block).
The curvature correction factor K
c
was obtained by
means of an equation that incorporated the relative
eccentricity (e
0Ed
/h) and the geometric slenderness (º
g
)
from the results of the numerical simulation. Thus, for
a particular axial force on the column N
Ed,i
(Fig. 3), the
curvature correction factor K
c
is obtained by perform-
ing sequentially the following steps.
(a) First, the second-order eccentricity is obtained
e
2, i
ðÞ
SN
¼
( M
Ed, i
)
NS
( M
0Ed, i
)
NS
N
Ed, i
(10)
where (M
Ed,i
)
NS
is the ultimate bending moment of
the cross-section for an axial force N
Ed,i
computed
from the numerical simulation (NS); (M
0Ed,i
)
NS
is
the first-order ultimate bending moment of the
support for an axial load N
Ed,i
computed from the
numerical simulation (NS)
(b) This second-order eccentricity allows the nominal
curvature of the section to be computed using
equation (4)
1
r
NS
¼
10 e
2, i
ðÞ
NS
l
2
0
(11)
(c) Finally, the curvature correction factor is obtained
by solving equation (6)
1
1/r
0
h/2
i
s
å
yd
å
cu2
h
Fig. 2. Base curvature 1/r
0
N
uc
M
(M
Ed,i
)
NS
Interaction diagram of the
cross section (ë
g
5 0)
Interaction diagram of
the column (ë
g
. 0)
l
o
e
0
N
Ed,i
N
Ed,i
N
(M
0Ed,i
)
NS
N
Ed,i
Ä
Fig. 3. Interaction diagram between the support and the section
Table 2. Ultimate strain of concrete under axial load and bending
f
ck
:MPa
12 16 20 25 30 35 40 45 50 55 60 70 80 90
cu2
(‰) 3
.
53
.
12
.
92
.
72
.
62
.
6
Design method for slender columns
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K
c
ðÞ
NS
¼
1= r
ðÞ
NS
1= r
0
(12)
where 1/r
0
is the base curvature computed from equa-
tion (7).
As an example, Fig. 4 shows the curvature correction
factor (K
c
)
NS
obtained through the numerical sim-
ulation, in terms of the first-order relative eccen-
tricity [e
0Ed
/h ¼ [( M
0Ed,i
)
NS
/N
Ed,i
)/h] and the geometric
slenderness (º
g
¼ l
o
/h), for a square column with equal
reinforcement at the four corners, a mechanical reinfor-
cement ratio (ø) equal to 0
.
50 and a strength of the
concrete of 30 MPa.
The correction factor K
c
depends on the relative
eccentricity e
0Ed
/h, as can be inferred from Fig. 4, and
its value concurs noticeably when the relative eccentri-
city is equal to 0
.
50 for any geometric slenderness (º
g
).
In this paper, this point is termed 'pivot correction
factor' (K
cp
) and defines the border between two differ-
ent equations for K
c
in terms of e
0Ed
/h: the first one
being parabolic (for e
0Ed
/h < 0
.
50) and the second one
is linear (for e
0Ed
/h . 0
.
50).
Although the correction factor K
c
depends appreci-
ably on the mechanical reinforcement ratio (ø), the
proposed equation was formulated independently of this
parameter in order to simplify the application of the
method. The accuracy will be demonstrated later on in
this paper.
An upper envelope parabola (Fig. 5) was adjusted
from the results of the numerical simulation in order to
obtain the first equation of K
c
(for e
0Ed
/h < 0
.
50). This
parabolic equation has an ordinate in the origin
K
c
¼ 0
.
5 and a value of K
c
¼ 1
.
05 for e
0Ed
/h ¼ 0
.
50.
Therefore, the equation of the parabola is
e
0Ed
= h < 0
:
50
K
c
¼ 2
:
2 e
0Ed
= h 0
:
50
ðÞ
2
þ1
:
05 (13)
The next branch of K
c
(for e
0Ed
/h . 0
.
50) was as-
sumed to be linear, crossing the point K
c
¼ 1
.
05 for
e
0Ed
/h ¼ 0
.
50 and having a slope that varies in terms of
º
g
. It was observed that the slope of the straight line
decreases if the geometrical slenderness increases.
Fitting the linear equation to the numerical simula-
tion results produces the following equation
e
0Ed
= h . 0
:
50
K
c
¼ (1
:
15 º
g
=30) e
0Ed
= h 0
:
50
ðÞ
þ 1
:
05 , 2
:
5 (14)
The value of K
c
is stopped at 2
.
5 to prevent high
values of e
0Ed
/h from producing extremely high values
of this equation.
Nominal curvature (1/r) of a column for axial loads
and uniaxial bending under sustained loads
As is known, for the case of sustained loads, the
second-order effects are increased when the strain ow-
ing to creep grows. Hence, the nominal curvature of
the section is increased. For this reason, the nominal
curvature (1/r) obtained for short-term loads (equation
(6)) is augmented with a correction factor K
j
in order
to take into account the long-term effects.
1
r
¼ K
j
K
c
1
r
0
(15)
Figure 6 presents the values of the total curvature
correction factor (K
j
.
K
c
) computed from the numer-
ical simulation as regards the relative eccentricity e
0Ed
/
h for three different creep coefficients j (0, 1 and 3)
and for three values of the geometric slenderness º
g
(10, 20 and 30). From this figure it can be inferred that
the values of K
c
.
K
j
for j ¼ 1 and j ¼ 3 are appreci-
ably parallel to those obtained for j ¼ 0 (instantaneous
load) with a parallelism factor that is independent of
e
0Ed
/h. This factor is lower when the geometric slender-
ness (º
g
) increases. This behaviour indicates that as
slenderness grows, the second-order effects become in-
creasingly more dependent on the geometry of the sup-
port itself (geometric nonlinearity) than on the lower or
higher deformability of the column owing to creep
(material nonlinearity).
2·5
2·25
2
1·75
1·5
1·25
1
0·75
0·5
0·25
0
Equal reinforcement at four corners
f
c
5 30 MPa; ù 5 0·50
(K
c
)
NS
0 0·25 0·5 0·75 1 1·25 1·5 1·75 2
e
0Ed
/h
ë
g
5 10
ë
g
5 15
ë
g
5 20
ë
g
5 25
ë
g
5 30
Fig. 4. Curvature correction factor K
c
for short-term loads
0
0·2
0·4
0·6
0·8
1
1·2
Envelope curve
used for the
proposed method
0 0·2 0·4 0·50·30·1
(K
c
)
NS
e
0Ed
/
h
Fig. 5. Correction factor K
c
for relative eccentricities lower
than 0
.
50
Bonet et al.
8
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The function of K
j
is obtained as a least square
adjustment from the numerical results, such as
K
j
¼ 1 þ 5 j=º
g
(16)
The parabolic branch of K
j
.
K
c
could be improved by
taking into consideration the fact that the short-term
values of K
c
are not perfectly parallel. The curvature of
the parabola increases with the creep coefficient j.
Thereby, equation (13) for calculating K
c
was modified
to take this effect into account
e
0Ed
= h < 0
:
50
K
c
¼ 2
:
2 þ j= 3
:
75
ðÞ
e
0Ed
= h 0
:
50
ðÞ
2
þ1
:
05 (17)
For the linear branch of K
j
.
K
c
the blocking value of
K
c
for instantaneous loads needs to be a function of j
e
0Ed
= h . 0
:
50
K
c
¼ (1
:
15 º
g
=30) e
0Ed
= h 0
:
50
ðÞ
þ 1
:
05 , (2
:
5 þ 0
:
8 j ) (18)
Also, the 'proposed' total nominal curvature correc-
tion factor ( K
j
.
K
c
) is presented in Fig. 6. In general the
proposal is slightly higher than the factor (K
j
.
K
c
) ob-
tained through the numerical simulation.
For the case where the permanent load applied to the
column is different to the total load, the creep coeffi-
cient (j) from equations (16) to (18) will be replaced
by the effective creep ratio (j
ef
).
This coefficient is obtained as the product between
the creep coefficient times the ratio between the first-
order bending moment in quasi-permanent load comb-
ination, SLS (M
0Eqp
) and the first-order bending
moment in design load combination, ULS (M
0Ed
).
j
ef
¼ j
M
0Eqp
M
0Ed
(19)
Nominal curvature (1/r) of a column subjected to axial
load and biaxial bending
It is important to note that if the support is subjected
to axial loads and biaxial bending, the second-order
Numerical simulation ö 5 3
Proposed method ö 5 3
Reinforcement equally
distributed at the four faces
f
c
5 30 MPa; ù 5 0·50
0
1
2
3
4
5
Kj· K
c
ë
g
5 10
0
1
2
3
4
5
0
1
2
3
4
5
0 0·25 0·5 0·75 1 1·25 1·5 1·75
e
0Ed
/h
Kj· K
c
0 0·25 0·5 0·75 1 1·25 1·5 1·75
e
0Ed
/h
ë
g
5 20
Kj· K
c
0 0·25 0·5 0·75 1 1·25 1·5 1·75
e
0Ed
/h
ë
g
5 30
Numerical simulation ö 5 1
Numerical simulation ö 5 0
Proposed method ö 5 1
Proposed method ö 5 0
Fig. 6. Total nominal curvature correction factor K
c
K
j
under sustained loads
Design method for slender columns
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eccentricity e
2
(equation (4)) is performed in accor-
dance with the bending plane of the first-order eccen-
tricity (Fig. 1).
The equation of the nominal curvature for axial loads
and uniaxial bending was expanded for the biaxial case
1
r
¼ K
j
K
c
1
r
0
¼ K
j
K
c
cu2
þ
yd
d
eq
(20)
where K
c
is the curvature correction factor
e
0Ed
= h
c
< 0
:
50
K
c
¼ 2
:
2 þ j
ef
=3
:
75
ðÞ
e
0Ed
= h
c
0
:
50
ðÞ
2
þ1
:
05
e
0Ed
= h
c
. 0
:
50
K
c
¼ (1
:
15 º
g
=30) e
0Ed
= h
c
0
:
50
ðÞ
þ 1
:
05 , (2
:
5 þ 0
:
8 j
ef
) (21)
K
j
is the correction factor to take into consideration
the increment in the nominal curvature owing to the
creep deformation
K
j
¼ 1 þ 5 j
ef
=º
g
(22)
e
0Ed
is first-order eccentricity, equal to the ratio be-
tween the vector modulus of the first-order bending
moment (M
0Ed
) and the factored axial load (N
Ed
)
e
0Ed
¼ M
0Ed
= N
Ed
M
0Ed
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
M
2
0Ed y
þ M
2
0Ed z
q
(23)
h
c
is critical dimension of the cross-section: the mini-
mum between the height and the width of the section,
min(b, h); º
g
is the geometric slenderness of the col-
umn
º
g
¼ l
0
= h
c
(24)
and d
eq
is the equivalent effective depth.
The equivalent effective depth of the cross-section is
linearly interpolated from the effective depths of the
section d
y
and d
z
with regard to the symmetry axes of
the section (Fig. 1).
d
eq
¼ d
z
þ d
y
1
ðÞ
(25)
where
d
z
¼ h=2 þ i
s z
d
y
¼ b=2 þ i
s y
i
sy
,i
sz
are the radii of gyration of the reinforcements
with respect to the coordinate axes of the section (Fig.
1 and Appendix 1) and is interpolation function.
In order to obtain an analytical equation to compute
the equivalent effective depth (d
eq
), the behaviour of a
support subjected to axial load and bending moment is
studied.
To clear the matter, Fig. 7(a) presents the dimension-
less interaction surface (taking into account the second-
order effects) of a rectangular column (h/b ¼ 2). The
mechanical reinforcement ratio (ø) is equal to 0
.
11 and
the concrete strength is 82 MPa. Fig. 7(b) displays the
dimensionless interaction diagram between the axial
load and the bending moment in the strong axis (
Ed
,
0Ed y
). Fig. 7(c) shows the dimensionless interaction
diagram (
0Ed y
,
0Edz
) for three levels of relative axial
load (
Ed
).
It can be inferred from Figs 7(a) and (b) that if the
column is subjected to axial load and uniaxial bending
with respect to the strong axis, the critical axial load of
the support (
cr
) is different if the deflection in the
weak axis is neglected or not. For the case, in which it
is neglected (braced column) the critical axial load
corresponds to the strong axis one (
cr,strong
). On the
other hand, if the column is unbraced and it is sub-
jected to axial load and uniaxial bending with respect
to the strong axis, the weak axis has a lot of influence
and the critical axial load corresponds to the weak axis
one (
cr,weak
).
The reduction of the strength capacity of the column
owing to the influence of the weak axis is important for
axial loads close to the critical (
cr,weak
). However, for
small levels of axial load, this reduction is insignificant
(Fig. 7(b)). This effect is higher as the applied axial load
(N
Ed
), the biaxial load level ( ), the ratio height/width
(h/b) and the slenderness (l
0
/h
c
) increase. Besides, it is
worth noting that for axial loads close to the critical
(
cr,weak
), the diagram (
0Ed y
,
0Ed z
) adopts a concave
shape. In general, any of these effects are considered in
the simplified methods proposed by different authors,
producing states on the unsafe side. Therefore, studying
the performance of the column (Fig. 7), the interpolation
function (equation (25)) must fulfil two conditions.
The first is that the equivalent effective depth d
eq
must tend towards the effective depth of the weak axis
(d
y
) when the first-order relative eccentricity (e
0Ed
/h
c
)
is close to zero because the behaviour of the columns is
strongly affected by the weak axis in this type of forces.
In other words, if e
0Ed
/h
c
tends to be zero, should be
0. The second condition is for case when the axial load
is zero N
Ed
¼ 0 (pure bending). If the eccentricity is
applied over the weak axis (z axis), is equal to one
(d
eq
¼ d
z
); and if the eccentricity is applied over the
strong axis (y axis), is zero (d
eq
¼ d
y
). The value of
will be enclosed between the two values in any
intermediate case.
With these conditions, the interpolation function is
obtained by the least square adjustment of the numer-
ical simulation results:
¼ cos
2
e
0Ed
= h
c
e
0Ed
= h
c
þ 10
(26)
where * is the relative biaxial bending moment angle
with respect to the strong axis
¼ arctan
M
0Ed z
h
M
0Ed y
b
(27)
Observe that if equation (25) is used for the cases of
Bonet et al.
10
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uniaxial bending, it should be noted whether or not the
support has neglected the bending in the transversal
plane. If this is case, the equivalent effective depth
(d
eq
) should be computed from the correspondent bend-
ing plane (d
y
or d
z
) and if it is not neglected equation
(25) will be valid.
Proposal of the simplified method
If all the previous factors are considered, and based
on the parametric study, the design bending moment of
the column (M
Ed
) can be obtained as the product of the
design axial load (N
Ed
) and the total design eccentricity
(e
Ed
). Such bending moment has the same direction as
the first-order bending moment applied at the ends of
the support.
M
Ed
¼ N
Ed
e
Ed
e
Ed
¼ e
0Ed
þ e
2
(28)
where
e
0Ed
is the vector modulus of the first-order eccentri-
city, e
2
is the second-order eccentricity
e
2
¼
1
r
l
2
0
10
(29)
1/r is the nominal curvature
0 0·02 0·04 0·06 0·08 0·1
0
0·01
0·02
0·03
0·04
0·05
0·06
0
0·02
0·04
0·06
0
0·02
0·04
0·06
0·08
0·1
0
0·2
0·4
0·6
0·8
1
Interaction diagram of the
braced
column in
uniaxial bending of the strong axis, when the
deflection in the weak axis is neglected
í
cr
5
strong
í
cr
5
weak
í
Ed
5
0·05
0 0·05 0·1
0
0·2
0·4
0·6
0·8
1
Braced column
Unbraced column
í
Ed
5
0·05
í
Ed
5
0·22
í
Ed
5
0·38
0·10 m
0·20 m
0·02 m
0·02 m
y
z
410
(a)
í
cr
5
weak
í
cr
5
strong
Interaction diagram of the
unbraced
column in
uniaxial bending of the strong axis, when the
deflection in the weak axis is
not
neglected
Column length (
l
0
)
5
3 m
ë
g
y
5
l
0
/
h
5
15;
ë
g
z
5
l
0
/
b
5
30
f
c
5
82 MPa;
f
y
5
558 MPa
í
Ed
5
0·22
í
Ed
5
0·38
í
Ed
5 N
Ed
/(A
c
·f
c
)
ì
0Ed
y
5
M
0Ed
y
/(
A
c
·f
c
·h
)
ì
0Ed
z
5
M
0Ed
z
/(
A
c
·f
c
·h
)
í
Ed
5 N
Ed
/(A
c
·f
c
)
ì
0Ed
y
5
M
0Ed
y
/(
A
c
·f
c
·h
)
ì
0Edz
5
M
0Ed
/(
A
c
·
f
c
b)
ì
0Ed
y
5
M
0Ed
y
/(
A
c
·
f
c
h)
(b) (c)
Fig. 7. Structural behaviour of the support subject to axial load and biaxial bending. (a) Dimensionless interaction surface with
second-order effects. (b) Dimensionless interaction diagram (
Ed
,
0Edy
) of the column in uniaxial bending of the strong axis. (c)
Dimensionless interaction diagram (
0Ed y
,
0Ed z
) of the unbraced column in biaxial bending
Design method for slender columns
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1
r
¼ K
j
K
c
1
r
0
(30)
1/r
0
is the base curvature
1
r
0
¼
cu2
þ
yd
d
eq
(31)
K
c
is the correction factor of the curvature
e
0Ed
= h
c
< 0
:
50
K
c
¼ 2
:
2 þ j
ef
=3
:
75
ðÞ
e
0Ed
= h
c
0
:
50
ðÞ
2
þ1
:
05
e
0Ed
= h
c
. 0
:
50
K
c
¼ (1
:
15 º
g
=30) e
0Ed
= h
c
0
:
50
ðÞ
þ 1
:
05 , (2
:
5 þ 0
:
8 j
ef
) (32)
K
j
is the correction factor of the curvature for taking
into account the long-term effects
K
j
¼ 1 þ 5 j
ef
=º
g
(33)
The equivalent effective depth of the section (d
eq
)
critical dimension of the cross-section (h
c
) can be
obtained from the following equations.
(a) For a 'braced' column subjected to axial load and
uniaxial bending moment with respect to the strong
axis
d
eq
¼ h=2 þ i
s y
h
c
¼ h (34)
(b) For an 'unbraced' column subjected to axial load
and uniaxial or biaxial bending moment
d
eq
¼ d
z
þ d
y
1
ðÞ
h
c
¼ min( h, b) (35)
where
¼ cos
2
e
0Ed
= h
c
e
0Ed
= h
c
þ 10
(36)
¼ arctan
M
0Ed z
h
M
0Ed y
b
(37)
Verification of the proposed method
The simplifications that were adopted make it neces-
sary to analyse the accuracy obtained by using the
proposed equation with respect to 371 experimental
results from the literature. They are detailed in Table 3.
The experimental tests have pinned–pinned boundary
conditions. These included cases of both uniaxial and
biaxial bending moment with axial loads, but always
with a rectangular section and doubly symmetric rein-
forcement.
The accuracy of the proposed method is estimated
from the ratio between the ultimate axial load from the
proposed simplified method (N
s
) and the experimental
tests (N
t
) for the same first-order eccentricity applied at
both ends.
Table 3. Verification of the proposed method by comparison with experimental tests
Short-term loads Sustained loads Total
No.
m
VC
m
´
ax
m
´
ın
No.
m
VC
m
´
ax
m
´
ın
No.
m
VC
m
´
ax
m
´
ın
Sarker et al .
22
12 0
.
80 0
.
11 1
.
18 0
.
70 — — — — — 12 0
.
80 0
.
11 1
.
18 0
.
70
Kim and Lee
23
16 0
.
89 0
.
16 1
.
13 0
.
74 — — — — — 16 0
.
89 0
.
16 1
.
13 0
.
74
Claeson and Gylltoft
24
20
.
99 0
.
06 1
.
03 0
.
95 2 0
.
89 0
.
01 0
.
90 0
.
88 4 0
.
94 0
.
07 1
.
18 0
.
88
Claeson and Gylltoft
25
12 0
.
87 0
.
09 1
.
15 0
.
75 — — — — — 12 0
.
87 0
.
09 1
.
15 0
.
75
Foster and Attard
26
54 0
.
83 0
.
11 1
.
15 0
.
62 — — — — — 54 0
.
83 0
.
11 1
.
15 0
.
62
Lloyd and Rangan
27
36 0
.
93 0
.
11 1
.
15 0
.
72 — — — — — 36 0
.
93 0
.
11 1
.
15 0
.
72
Kim and Yang
28
30 0
.
91 0
.
09 1
.
08 0
.
79 — — — — — 30 0
.
91 0
.
09 1
.
08 0
.
79
Hsu et al.
29
70
.
73 0
.
10 1
.
00 0
.
64 — — — — — 7 0
.
73 0
.
10 1
.
00 0
.
64
Tsao and Hsu
30
60
.
88 0
.
11 1
.
01 0
.
76 — — — — — 6 0
.
88 0
.
11 1
.
01 0
.
76
Wang and Hsu
31
80
.
93 0
.
06 1
.
02 0
.
86 — — — — — 8 0
.
93 0
.
06 1
.
02 0
.
86
CEB
21
———— 80
.
92 0
.
14 1
.
11 0
.
76 8 0
.
92 0
.
14 1
.
11 0
.
76
Mavichak and Furlong
32
90
.
97 0
.
11 1
.
09 0
.
81 — — — — — 9 0
.
97 0
.
11 1
.
09 0
.
81
Wu
33
11 0
.
78 0
.
06 0
.
89 0
.
71 — — — — — 11 0
.
78 0
.
06 0
.
89 0
.
71
Goyal and Jackson
34
26 0
.
99 0
.
13 1
.
18 0
.
76 20 0
.
96 0
.
13 1
.
14 0
.
69 46 0
.
98 0
.
13 1
.
18 0
.
69
Drysdale and Huggins
35
26 0
.
74 0
.
11 0
.
94 0
.
66 31 0
.
77 0
.
12 0
.
98 0
.
63 57 0
.
76 0
.
12 0
.
98 0
.
63
Breen and Ferguson
36
30
.
82 0
.
22 0
.
96 0
.
61 — — — — — 3 0
.
82 0
.
22 0
.
96 0
.
61
Green and Breen
37
————— 21
.
03 0
.
03 1
.
05 1
.
01 2 1
.
03 0
.
03 1
.
05 1
.
01
Chang and Ferguson
38
60
.
75 0
.
15 1
.
18 0
.
61 — — — — — 6 0
.
75 0
.
15 1
.
18 0
.
61
Viest et al.
39
15 0
.
94 0
.
11 1
.
13 0
.
73 29 0
.
95 0
.
10 1
.
13 0
.
67 44 0
.
94 0
.
10 1
.
13 0
.
67
279 0
.
87 0
.
14 1
.
18 0
.
61 92 0
.
89 0
.
15 1
.
14 0
.
63 371 0
.
88 0
.
14 1
.
18 0
.
61
(*) The value of the variation coefficient is not representative owing to the small number of tests.
m
: average ratio; VC.: variation coefficient;
max
: maximum ratio;
m
´
ın
: minimum ratio.
Bonet et al.
12
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¼
N
s
N
t
(38)
To calculate the ultimate bending moment of the
column cross section, the parabola–rectangle defined in
the draft of EC-2
15
(Fig. 8) was selected.
The characteristic ( f
ck
and f
yk
) and design ( f
cd
and
f
yd
) strengths of concrete and steel are taken as being
similar to those in the experimental tests ( f
c
and f
y
),
respectively. Table 4 shows the variation of the para-
meters studied in the experiments.
Table 3 shows separate views of the accuracy
achieved with the proposed method for both short-term
loads and sustained loads. The mean of all the tests is
also included. The average ratio for short-time loads is
0
.
87 with a variation coefficient of 0
.
14, whereas for
sustained loads the ratio is 0
.
89 with a variation of
0
.
15.
Finally, for all the experiments, regardless of the
load, the average ratio was 0
.
88 with a variation coeffi-
cient of 0
.
14.
Figure 9 shows the variation of (obtained for all
the cases) in terms of the most important parameters
drawing a trend line in each graph. The selected vari-
ables were: compressive strength ( f
c
), steel yielding
stress ( f
y
), mechanical reinforcement ratio (ø), geo-
metric slenderness of the column (º
g
), relative axial
load (
Ed
), relative biaxial bending angle ( *), effective
creep ratio (j
ef
), and the auxiliary parameter Ł, the
analytical expression of which is defined below
Ł( rad ) ¼ tan
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
M
0Ed y
= h
2
þ M
0Ed z
= b
ðÞ
2
q
N
Ed
2
4
3
5
(39)
The parameter Ł allows us to analyse the degree of
accuracy in terms of the relative eccentricity applied at
the section (e
0Ed
/h
c
). It is delimited by Ł equal to /2
when N
Ed
¼ 0 (pure bending) and Ł equal to 0 when
the first-order relative eccentricity (e
0Ed
/h
c
) is zero
(pure compression).
The trend lines of the graphs in Fig. 9 are horizontal
or with a slight slope (increasing or decreasing). This
means that the proposed method correctly detects the
variation of such parameters with a reasonable devia-
tion.
A comparison with the experimental tests in Table 3
ó
c
0·85 f
cd
where n
5
1·4
1
23·4 [(90
2
f
ck
)]/100]
4
<
2·0
(b) For
å
cu
<
å
c
<
å
cu2
(a) For 0
<
å
c
<
å
c2
å
c2
å
cu2
å
c
ó
c
5
0·85
f
cd
· 1
2
1
2
å
c
å
c2
n
ó
c
5
0·85 f
cd
å
c2
(
)
5
2·0
1
0·085 (f
ck
2
50)
0·53
>
2·0
å
cu2
(
)
5
2·6
1
35 [(90
2
f
ck
)/100]
4
<
3·5
Fig. 8. Parabola–rectangle diagram
Table 4. Parameter variation in the experimental tests
Parameter Range
Compressive concrete strength [ f
c:
] 10
.
76–107 MPa
Steel strength [ f
y
] 298
.
55–684 MPa
Mechanical reinforcement ratio [ø] 0
.
07–1
.
42
Volumetric reinforcement ratio [r
g
] 0
.
01–0
.
05
Type of section Rectangular or square
Reinforcement distribution Doubly symmetric
Column geometric slenderness [º
g
] 3–40
Height/width ratio [h/b] 1–2
Relative axial load [
Ed
] 0
.
04–1
.
20
Relative eccentricity [Ł] 0
.
05–1
.
05
Relative biaxial angle [ *] 0–90 8
Creep coefficient [j] 0
.
32–3
.
29
Efficient creep ratio [ j
ef
] 0
.
32–2
.
83
Ratio between the first-order bending moment in quasi-permanent load combination (M
0Eqp
) and the first-order
bending moment in design load combination, ULS (M
0Ed
):
M
0Eqp
M
0Ed
0
.
42–1
.
00
Design method for slender columns
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applying the method of the draft EC-2
15
was carried
out to evaluate the degree of accuracy reached with the
proposed method.
The function of the second-order eccentricity pro-
posed by the draft of EC-2
15
for the case where the
supports are subjected to axial load and uniaxial bend-
ing is
e
2
¼
1
r
l
2
0
c
¼ K
r
K
j
1
r
0
l
2
0
c
(40)
0·5
0·6
0·7
0·8
0·9
1
1·1
1·2
1·3
1·4
f
c
: MPa
(a)
0·5
0·6
0·7
0·8
0·9
1
1·1
1·2
1·3
1·4
0·5
0·6
0·7
0·8
0·9
1
1·1
1·2
1·3
1·4
0·5
0·6
0·7
0·8
0·9
1
1·1
1·2
1·3
1·4
0·5
0·6
0·7
0·8
0·9
1
1·1
1·2
1·3
1·4
0·5
0·6
0·7
0·8
0·9
1
1·1
1·2
1·3
1·4
0·5
0·6
0·7
0·8
0·9
1
1·1
1·2
1·3
1·4
j
ef
(h)
0·5
0·6
0·7
0·8
0·9
1
1·1
1·2
1·3
1·4
0 102030405060708090100110
îî
0·2 0·4 0·6 0·8 1 1·20
ù
5
(
A
s
·
f
y
)/(
b
·
h
·
f
c
)
(c)
í
Ed
5
N
Ed
/(
b
·
h
·
f
c
)
(e)
0·2 0·4 0·6 0·8 1 1·20
0·2 0·4 0·6 0·8 1 1·20 1·4
î
0 153045607590
â
*(°)
5
arctan[(M
0Edz
·h)/(M
0Edy
·b)]
(g)
î
200 300 400 500 600 700 800
f
y
: MPa
(b)
îî
î
0 10203040
ë
g
(d)
è
(rad)
5
[tan
21
([
M
0Edy
/
h
)
2
1
(
M
0Edz
/
b
)
2
]
0·5
/
N
Ed
)]
(f)
î
0 0·5 1 1·5 2 2·5 3
Fig. 9. Comparison of the proposed method with the experimental results in terms of: (a) concrete trength; (b) steel strength; (c)
reinforcement ratio; (d) slenderness; (e) relative axial load; ( f) bending moment angle; (g) relative biaxial bending moment; (h)
effective creep ratio
Bonet et al.
14
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where K
r
is the correction factor, which depends on the
axial level
K
r
¼
n
u
n
n
u
n
bal
< 1 (41)
n is the relative axial load [N
Ed
/(A
c
f
cd
)], N
Ed
is the
design axial load of the support, A
c
is the area of
concrete cross section, f
cd
is the design strength of con-
crete, n
u
is the ultimate relative axial load of the sec-
tion under pure compression [N
uc
/(A
c
f
cd
)] (Fig. 3),
n
bal
is the value of n at maximum moment resistance,
K
j
is the correction factor, which takes into account
the creep effect
K
j
¼ 1 þ j
ef
> 1 (42)
being
¼ 0
:
35 þ f
ck
=200 º =150
f
ck
is the characteristic strength of concrete (in MPa), º
is the mechanical slenderness of the support
º ¼ l
0
= i (43)
i is the radius of gyration of the 'gross' section, 1/r
0
is
the base curvature
1= r
0
¼
yd
= 0
:
45 d
ðÞ
(44)
where
yd
¼ f
yd
= E
s
, f
yd
is the design yielding stress of
reinforcement, E
s
is the elastic modulus of the reinfor-
cing bars and d is the effective depth. If all reinforce-
ment is not concentrated on opposite sides, but part of
it is distributed parallel to the plane of bending, d is
defined as
d ¼ h=2 þ i
s
(45)
where h is the height of the section (parallel to the
bending plane), i
s
is the radius of gyration of the
reinforcing bars, c is a factor that takes into considera-
tion the curvature distribution along the column. For a
sinusoidal curvature distribution, a value of 10 (
2
)is
adopted.
If the column is subjected to biaxial bending and
axial loads, the method described by equation (1) is
applied. For the computation of the ultimate bending
moments of the section, the parabola–rectangle diagram
defined by EC-2
15
(Fig. 8) is applied. It is identical
with the one considered in the verification of the meth-
od proposed in the current paper.
Table 5 presents a comparison between the results
obtained by applying the proposed method and the one
from EC-2
15
with respect to experimental results.
In general, the proposed method achieves an average
ratio closer to one on the safe side, with a lower devia-
tion coefficient. It is important to notice that the pro-
posed method presents an essential improvement for
sustained loads, both for uniaxial and biaxial bending,
and also for short-term loads and biaxial bending.
Only for the case of uniaxial bending and short-term
loads does the method from EC-2
15
present similar
results to those obtained by the proposed method.
Example
In order to illustrate the practical application of the
proposed method, the longitudinal reinforcement of an
unbraced column is calculated. The column has a buck-
ling length of 4 m and it is subjected to constant forces
along the length of the element corresponding to the
ultimate limit state for the permanent or variable state.
These are N
Ed
¼ 2300 kN, M
0Ed y
¼ 60 kN m and
M
0Ed z
¼ 45 kN m. The cross section is presented in
Fig. 10. The mechanical properties of the materials are
f
ck
¼ 80 MPa and f
yk
¼ 500 MPa and E
s
¼ 200 000
MPa with a normal level of quality control. The effec-
tive creep ratio (j
ef
) is equal to 1
.
2.
The size of the reinforcement is obtained by follow-
ing the steps explained in previous sections using the
basic hypothesis from EC-2
15
to compute the ultimate
bending moments.
Initially, the following parameters are computed
Table 5. Comparison of the obtained results from the proposed method and EC-2 with experimental results; classified by the
type of external load
Type of external
force
Type of time
load
Method No
:
of tests
m
VC
min
max
Uniaxial bending Short-term Proposed 194 0
.
90 0
.
13 0
.
61 1
.
18
EC2 (2004) 0
.
95 0
.
15 0
.
61 1
.
29
Sustained Proposed 65 0
.
94 0
.
12 0
.
67 1
.
14
EC2 (2004) 1
.
11 0
.
17 0
.
59 1
.
62
Biaxial bending Short-term Proposed 85 0
.
82 0
.
15 0
.
64 1
.
13
EC2 (2004) 0
.
67 0
.
24 0
.
40 0
.
98
Sustained Proposed 27 (*) 0
.
75 0
.
11 0
.
63 0
.
91
EC2 (2004) 0
.
67 0
.
14 0
.
54 0
.
85
Total Proposed 371 0
.
88 0
.
14 0
.
61 1
.
18
EC2 (2004) 0
.
89 0
.
25 0
.
40 1
.
62
(*) All the tests are from the same authors (Drysdale and Huggins
35
).
m
: average ratio; VC: variation coefficient;
max
: maximum ratio;
m
´
ın
: min ratio.
Design method for slender columns
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f
cd
¼ f
ck
=ª
c
¼ 80=1
:
5 ¼ 53
:
3 MPa
f
yd
¼ f
yk
=ª
s
¼ 500 =1
:
15 ¼ 434
:
78 MPa
h
c
¼ min( h, b) ¼ min(0
:
25, 0
:
40) ¼ 0
:
25 m
º
g
¼ l
0
= h
c
¼ 4=0
:
25 ¼ 16
yd
¼ f
yd
= E
s
¼ 434
:
78=200 000 ¼ 0
:
00217
cu2
(‰) ¼ 2
:
6 þ 35 90 f
ck
ðÞ
=10
4
¼ 2
:
6 þ 35 90 80
ðÞ
=10
4
¼ 2
:
6 , 3
:
5
The second-order eccentricity is obtained using equa-
tion (29). Previously, the following computations need
to be performed.
(a) First-order relative eccentricity (e
0Ed
/h
c
)
e
0Ed y
¼
M
0Ed z
N
d
¼
45
2300
¼ 0
:
0195 m
e
0Ed z
¼
M
0Ed y
N
d
¼
60
2300
¼ 0
:
026 m
e
0Ed
= h
c
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
e
2
0Ed y
þ e
2
0Ed z
q
= h
c
¼ 0
:
0326=0
:
25 ¼ 0
:
1304
(b) Correction factor K
c
(equation (32)) for e
0 Ed
/h
c
<
0
.
5
K
c
¼ 2
:
2 þ j
ef
=3
:
75
ðÞ
e
0Ed
= h
c
0
:
50
ðÞ
2
þ1
:
05
¼ 2
:
2 þ 1
:
2=3
:
75
ðÞ
0
:
1304 0
:
50
ðÞ
2
þ 1
:
05 ¼ 0
:
793
(c) Correction factor K
j
owing to sustained loads
(equation (33))
K
j
¼ 1 þ 5 j
ef
=º
g
¼ 1 þ 5(1
:
2=16
½
¼ 1
:
375
(d) Radii of gyration of the reinforcements with re-
spect to the coordinate axes of the section (Appen-
dix 1).
i
s z
¼
d d 9
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
4 n
z
( n
z
þ 2)
3 ( n
z
þ 1) (4 þ 2 n
y
þ 2 n
z
)
s
¼
0
:
35 0
:
05
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
4 3 (3 þ 2)
3 (3 þ 1) (4 þ 2 1 þ 2 3)
s
¼ 0
:
11456 m
i
s y
¼
0
:
20 0
:
05
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
4 1 (1 þ 2)
3 (1 þ 1) (4 þ 2 3 þ 2 1)
s
¼ 0
:
06846 m
(e) Relative biaxial bending moment * (equation
(37))
¼ tan
1
M
0Ed z
h
M
0Ed y
b
¼ tan
1
45 3 0
:
40
60 3 0
:
25
¼ 50
:
19
( f ) Interpolation coefficient (equation (36))
¼ cos
2
e
0Ed
= h
c
e
0Ed
= h
c
þ 10
¼ cos
2
50
:
19
0
:
1304
0
:
1304 þ 10
¼ 0
:
00527
(g) The equivalent effective depth is computed from
the following equation (equation (35))
d
eq
¼ d
z
þ d
y
1
ðÞ
¼ (0
:
3146) (0
:
0527) þ 0
:
1935 (1 0
:
0527)
¼ 0
:
1941 m
where
d
z
¼ h=2 þ i
s z
¼ 0
:
40=2 þ 0
:
11456 ¼ 0
:
3146 m
d
y
¼ b=2 þ i
s y
¼ 0
:
25=2 þ 0
:
06846 ¼ 0
:
1935 m
(h) The nominal curvature (equation (30)) adopts this
value
1
r
¼ K
j
K
c
cu2
þ
yd
d
eq
¼ (0
:
793) (1
:
375)
0
:
00217 þ 0
:
0026
0
:
1941
¼ 0
:
02684 m
1
(i) The second-order eccentricity (equation (29)) is
e
2
¼
1
r
l
2
0
10
¼ 0
:
02684
4
2
10
¼ 0
:
0435 m
( j) Finally, the total design eccentricity (equation (28))
is equal to
e
Ed
¼ e
0Ed
þ e
2
¼ 0
:
0326 þ 0
:
0435 ¼ 0
:
0761 m
being
e
0Ed
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
e
2
0Ed y
þ e
2
0Ed z
q
¼ 0
:
0326 m
The vector modulus of the total design bending mo-
0·40
0·25
0·05
0·05
z
y
12ö?
In metres
Fig. 10. Example cross section of the support
Bonet et al.
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ment (M
Ed
), according to the first-order bending mo-
ment plane with regard to the strong axis (equation
(28)), is
M
Ed
¼ N
Ed
e
Ed
¼ 2300 3 0
:
0761 ¼ 175
:
07 kN m
where
¼ tan
1
M
0Ed z
M
0Ed y
¼ tan
1
45
60
¼ 36
:
878
Consequently, the design forces are
N
Ed
¼ 2300 kN; M
Ed y
¼ 140
:
06 kN m;
M
Ed z
¼ 105
:
04 kN m
From these forces, the longitudinal reinforcement that
is needed is calculated in accordance with the distribu-
tion indicated in Fig. 10. By so doing, the required area
of reinforcement is found to be equal to 22
.
54 cm
2
(12
bars with diameter ¼ 16 mm).
Conclusions
The present paper proposes a simplified method for
designing slender rectangular reinforced concrete col-
umns with doubly symmetric reinforcement subjected
to combined axial loads and biaxial bending that is
valid for short-time and sustained loads, and for both
normal- and high-strength concretes. The method is
only valid for columns with equal effective buckling
lengths in the two principal bending planes. It is an
extension for biaxial bending of the column-model
method.
A new equation is presented to obtain the nominal
curvature (1/r) of the critical section of columns with
doubly symmetric reinforcement subjected to combined
axial loads and uniaxial bending.
The proposed formulation for biaxial bending is an
extension of the general nominal curvature (1/r) equa-
tion for uniaxial bending obtained by calculating the
equivalent effective depth of the column cross section.
This formulation includes the existing interaction be-
tween both flexure axes and the particular case of the
axial load and single curvature. The effect of braced
structures is taken into account in the behaviour of the
column subjected to an axial load and uniaxial bending
with respect to the strong axis.
The method was compared with 371 experimental
tests and it proved to be accurate enough for its practi-
cal application.
The accuracy of the proposed method was compared
with the equations proposed by EC-2,
15
and a notice-
able improvement was accomplished. It is important to
highlight that this improvement is more relevant for
sustained loads and biaxial bending. The draft of EC-2
has a different method for uniaxial bending than for
biaxial bending, while the proposed method has a uni-
fied formulation.
Unlike other simplified methods (i.e. EC-2), the pro-
posed method can be directly applied to design pur-
poses because it does not require any iterative process,
since it is independent of the mechanical reinforcement
ratio.
Acknowledgements
The authors wish to express their sincere gratitude to
the Spanish Ministerio de Ciencia y Tecnologı´a for help
provided through project MAT2002-02461, and also to
the Ministerio de Fomento (BOE 13/12/2002).
Design method for slender columns
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Appendix 1. Radii of gyration
The equations of the radius of gyration with respect to the horizontal axis of the most common cases are presented in the table
below .
Reinforcement distribution Radius of gyration(i
s
)
Equal at opposite faces
A
A
dd¢
( d d 9 )
2
A
A
dd¢
( d d 9 )
ffiffiffiffiffi
12
p
Equal at the four faces
A A
A
A
dd¢
( d d 9 )
ffiffiffiffiffi
16
p
Uniformly distributed
b¢
¢
h
¢
bA
¢
h
¢
bA
A
A
h¢ 5 dd¢
( d d 9 )
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3 b9 þ h9
12 ( b9 þ h9 )
r
Doubly symmetric (*)
( d d 9 )
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
4 n
z
( n
z
þ 2)
3 ( n
z
þ 1) (4 þ 2 n
y
þ 2 n
z
)
s
n
y
n
z
dd
¢
where
n
y
, n
z ¼
number of bars at the faces of the section
General. Doubly symmetric
n
y
n
z
A
se
A
sy
A
sz
dd
¢
( d d 9 )
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
4 n
z
ª
z
( n
z
þ 2)
3 ( n
z
þ 1)
s
where
n
y
,n
z
¼ number of bars at the faces of the section
ª
z
¼ A
s y
/A
s
A
s
¼ 4 A
se
+2 n
y
A
s y
+2 n
z
A
s z
A
se,
A
s y
,A
s z
represent area of one of the bars located at the corners
or at the faces of the section
(*) The obtained expression assumes that all the bars have the same diameter.
Bonet et al.
18
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n. Instruccio
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n Estructural. (EHE), Ministerio de Fomento, Madrid, 1999.
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Discussion contributions on this paper should reach the editor by
1 August 2007
Design method for slender columns
Magazine of Concrete Research, 2007, 59, No. 1 19
... Theoretical analysis of slender bi-axial bending columns should be always compared with the experimental research. The literature contains some papers in which theoretically analysed load-bearing capacity and deformability of columns and calculation results are compared with author's own experimental research: Afefy et al. [13], Kim and Lee [2], Laite et al. [14], Pallarés et al. [15], Ramamurthy [16] or with experimental research made by other authors: Afefy et al. [13], Ahmad and Weerakoon [17], Bonet et al. [18], Tikka and Mirza [19], Westerberg [20]. Proper estimation of second order effects in slender reinforced concrete columns is a complicated and difficult task. ...
... Proper estimation of second order effects in slender reinforced concrete columns is a complicated and difficult task. Despite of the fact that many authors in their publications, among others: Afefy et al. [13], Barros et al. [21,22], Bonet et al. [18], Bazant et al. [23], Khuntia and Ghosh [24,25] analysed second order effects, looking for the appropriate expressions to define stiffness or curvature of the deformed element, further analyses of this problem are necessary. ...
... Simplified approaches included in standards to determine second order effects are usually based on the nominal curvature method (NC)-EC2 [1] or nominal stiffness (NS)-EC2 [1] and ACI [2]. Most of the simplified approaches assume a separate calculation of second order effects for both the main column planes -EC2 [1] and ACI [2]; however, there are few papers estimating second order effects directly in the oblique deflection plane [4,5]. ...
... During the recent decades, many authors in their publications: Afefy et al. [32], Barros et al. [33], Bonet et al. [5], Bazant et al. [34], Diniz and Frangopol [35], Khuntia and Ghosh [36,37], MacGregor et al. [38,39], Mavichak and Furlong [40], Tikka and Mirza [41], Westerberg [42], analysed second order effects, looking for the appropriate expressions to define stiffness or curvature of the deformed, compressed element. ...
... Other authors present simplified design methods for slender columns under axial load and biaxial bending 3,[6][7][8][9][10] . Some papers deal with the simplified design procedures of columns according to Eurocode 2 recommending changes to increase the accuracy of the Moment magnification 11,12 or the Nominal Curvature 13,14 methods. ...
... i h d + = (12) where i bars is the radius of gyration of the rebars. (For the case shown in Fig. 12a d' = d.) ...
In this paper, expressions are developed which enable the designer to determine the load bearing capacity of concentrically loaded RC columns in a very simple manner. The 'reduction factor' (the ratio of the ultimate load of the column and that of the cross-section) is introduced. It is similar to that used for the calculation of steel, masonry and timber structures. The results are based on the second order analysis of RC columns taking also into account the eff ect of creep. The novelty of the paper is not the presented nonlinear solution of RC columns, rather the approximate 'back of the envelope' expressions, which are verifi ed for the entire practical parameter range by a numerical solution.
... There are several articles in the literature which deal with the design procedures of columns according to Eurocode 2 (Bonet et al. (2007), Bonet et al. (2004), Mirza and Lacroix (2002), Aschheim et al. (2007)), however none of these treats the centric loaded columns separately. Other parts of Eurocode contain simple methods, which can be used for the calculation of centric loaded columns. ...
- Bernát P Csuka-László
-
![László P. Kollár]()
The paper presents a very simple method for the design and analysis of centric loaded, symmetrically rein-forced concrete columns with rectangular or circular cross-sections. The concept of the "capacity reduction factor" (or "instability factor", "buckling coefficient") is introduced, which was applied for steel, timber and masonry columns in Eurocode 3, 5 and 6, respectively. The "capacity reduction factor" is determined on the basis of Eurocode 2. It is shown numerically that the method is always conservative and reasonably accurate. The usage of the method is demonstrated through numerical examples.
... Cross-sectional loads include second-order effects, recent interesting approaches can be used if those effects are not included [15,16]. ...
Recent advances in approaches to the design of reinforced concrete sections have culminated in a theorem of optimal (minimum) sectional reinforcement. This theorem is articulated on the basis of patterns observed in the optimal reinforcement of rectangular sections, obtained with a new approach for the analysis and design of reinforcement. Using the hypotheses for ultimate strength design sanctioned by ACI 318-05 (2008), the minimum total reinforcement area required to provide adequate resistance to axial load and moment is shown to occur for particular constraints on longitudinal reinforcement area or distributions of strain. These constraints are identified along with the solutions for minimum total reinforcement area. Optimal reinforcement may be selected from among the potential solutions identified by the theorem. An example illustrates the application of the theorem to the design of a reinforced concrete cross-section. Implications for teaching and practice of reinforced concrete design are discussed.
... parallel sides have equal reinforcement). Reinforcement configured in this way, with double symmetry, is often used for rectangular sections subjected to biaxial bending [20]. The consideration of double symmetry forces to a reduction in the number of variables, with this restriction, admissible solutions for strength design depend only on one variable, so the minimum reinforcement area can be identified on a 2D (two-axis) diagram, this point will be develop later on the paper. ...
The Reinforcement Sizing Diagram (RSD) approach to determining optimal reinforcement for reinforced concrete beam and column sections subjected to uniaxial bending is extended to the case of biaxial bending. Conventional constraints on the distribution of longitudinal reinforcement are relaxed, leading to an infinite number of reinforcement solutions, from which the optimal solution and a corresponding quasi-optimal pragmatic is determined. First, all possibilities of reinforcement arrangements are considered for a biaxial loading, including symmetric and non-symmetric configurations, subject to the constraint that the reinforcement is located in a single layer near the circumference of the section. This theoretical approach establishes the context for obtaining pragmatic distributions of reinforcement that are more suitable for construction, in which distributions having double symmetry are considered. This contrasts with conventional approaches for the design of column reinforcement, in which a predetermined distribution of longitudinal reinforcement is assumed, even though such a distribution generally is non-optimal in any given design. Column and wall sections that are subjected to uniaxial or biaxial loading may be designed using this method. The solutions are displayed using a biaxial RSD and can be obtained with relatively simple algorithms implemented in widely accessible software programs such as Mathematica® and Excel®. Several examples illustrate the method and the savings in reinforcement that can be obtained relative to conventional solutions. KeywordsUltimate strength design-Optimal reinforcement-Biaxial bending
- José Milton de Araújo
The purpose of this paper is to present a non-linear model for analysis and design of slender reinforced-concrete columns subjected to uniaxial and biaxial bending. This model considers both material and geometric non-linearities, as well as creep effects. The structural analysis is performed by the finite-element method associated with an iterative process to solve the system of non-linear equations. The column may have an arbitrary polygonal cross-section, including openings. Green's theorem is used to perform the integration at the level of the cross-sections, which is greatly simplified with the use of a new parabola-rectangle diagram proposed for concrete in compression. This new diagram provides the correct value of the tangent modulus of elasticity of concrete, allowing its use for non-linear analysis of slender columns. By changing the strain value corresponding to the maximum stress, it is possible to use a single stress-strain diagram for displacement calculation and rupture verification, which facilitates the design of slender columns. The accuracy of the method is demonstrated through the analysis of several columns tested experimentally by other authors.
- José Milton de Araújo
Usually, reinforced concrete design codes indicate only one simplified method for second order analysis of slender columns. The Eurocode 2 (EC2), on the other hand, adopts two simplified methods: one based on nominal stiffness and other based on nominal curvature. It would be desirable that both methods could provide similar solutions. However, this is not the case, as shown in this paper. On the contrary, the two EC2 simplified methods can provide very different results, leaving the engineer uncertain about which method he should use. The objective of this work is to compare these two simplified methods presenting the contradictions between them. Several experimental results available in the literature have been analysed and compared. The method based on the nominal curvature showed to be the most accurate; therefore, it is suggested to be used.
- Hyo-Gyoung Kwak
- Ji-Hyun Kwak
Nonlinear analyses are conducted to evaluate the ultimate resisting capacity of slender reinforced concrete (RC) columns subjected to an axial load with biaxial bending moments. Consideration is given to the geometric nonlinearities caused by the P–Δ effect and the long-term behavior of concrete and to the material nonlinearities caused by the cracking of concrete and the yielding of steel. In addition, the biaxial stress state in an RC section is simulated on the basis of a fiber model. Because of the complexity of Bresler's load contour method, which was introduced in the ACI 318 code, this paper introduces a new design approach to the construction of the failure surface of a slender RC column subjected to biaxial bending. Through a parametric study of slender RC columns, where consideration is given to the P–Δ effect and the time-dependent deformation of concrete, two regression formulas are proposed on the basis of the slenderness ratio and the creep deformation of concrete. Furthermore, the direct multiplication of the proposed formulas on the P–M interaction diagram for a short RC column subjected to axial force and a uniaxial bending moment enables a P–M interaction diagram to be generated for a slender RC column subjected to long-term axial force and biaxial bending moments. Correlation studies between analytical and experimental results are conducted with the objective of establishing the validity of the introduced numerical model. In addition, the ultimate resisting capacities calculated from the regression formula are compared with those obtained from rigorous nonlinear analyses and from the ACI formula, with the objective of establishing the relative efficiency of the proposed regression formula.
-
![Richard Furlong]()
- C.-T. Thomas Hsu
- S.A. Mirza
Columns with axial load causing biaxial bending are present in many different building structures. The provisions of ACI 318 Section 10.2 are the basis for traditional design aids that show section strength when moments act in a plane of symmetry. Strength analysis for biaxial bending is significantly more difficult, as moments are not applied in a plane of symmetry. Several methods of analyses that use traditional design aids are reviewed and the results are compared with data obtained from physical tests of normal strength concrete columns subjected to short-term axial loads and biaxial bending. Results indicate that any among the four different methods of cross-sectional analysis are equally suitable for design purposes. The value of three-dimensional interaction diagrams in the design process is discussed. Computer-based methods of analysis are also described and compared with test observations.
- C. Claeson
- K. Gylltoft
A test series examining the structural behavior of six slender reinforced concrete columns subjected to short-term and sustained loading is presented. The columns had cross sections 200 × 200 mm and were 4 m long. Concrete strengths used were 35 and 92 MPa. with a load eccentricity of 20 mm. Key parameters such as concrete strength, concrete and steel strains, cracking, midheight deflection, and loading rate were studied. The high-strength concrete (HSC) columns subjected to short-term loading displayed less ductility and more sudden failures than the normal strength concrete (NSC) columns. Furthermore, the tests conducted indicated that the structural behavior of the HSC is favorable under sustained loading, i.e., the HSC column exhibited less tendency to creep and could sustain the axial load without much increase in deformation for a longer period of time. An analysis based on a simplified stability analysis, using a stress-strain relation for concrete that includes creep, aging, and the confining effect of the stirrups was carried out. The model was shown to simulate the load-deflection curves satisfactorily for all of the concrete columns.
The results of a research program on the behavior and strength of high-strength concrete columns under eccentric compression are presented. Thirty-six columns were tested; the variables were column cross section, eccentricity of load, longitudinal reinforcement ratio, and concrete compressive strength. The columns were either 300 x 100 or 175 x 175 mm (12 x 4 or 7 x 7 in.) in cross section with an effective length of 1680 mm (66 in.). They were reinforced with either four or six deformed bars of 12 mm (0.5 in.) diameter and yield strength of 430 MPa (62 ksi). Concrete cylinder compressive strength at the time of testing was either 58, 92, or 97 MPa (8410, 13,340, or 14,065 psi). Eccentricity of load was varied in the range from 0.086 to 0.4 times the column depth and the rectangular specimens were loaded about the minor axis. Lateral reinforcement was provided by 4-mm (0.16-in.) closed ties with a minimum yield strength of 450 MPa at 60-mm (2.36-in.) spacing. A theory was developed to predict the load-deflection behavior and the failure load of high-strength concrete columns under eccentric compression. The theory is based on a simplified stability analysis and a stress-strain relation of high-strength concrete in compression. The average ratio of test failure load to predicted failure load is 1.13 with a coefficient of variation of 10 percent.
Reported in this paper are the test results for 68 eccentrically loaded conventional and high-strength concrete columns. The columns were 150 × 150 mm (5.91 × 5.91 in) at the mid-section and haunched at the ends to apply the eccentric loading and prevent boundary effects. Concrete strengths used were 40, 55, 75, and 90 M.B.A. (5800, 8000, 10,900, and 13,100 psi) with load eccentricities of 8, 20, and 50 mm (0.32, 0.79, and 1.97 in). The columns had either 2 or 4 percent longitudinal reinforcement and tie spacings of 30, 60, or 120 mm (1.81, 2.36, or 4.72 in). The ultimate strength of the columns is compared to the strength predictions based on the ACI 318-89 rectangular stress block parameters. The predictions compare reasonably well, although lower strengths than predicted occurred for some high-strength concrete specimens. Ductilities are calculated based on the area under the load versus average strain plus curvature times eccentricity relationship. This measure showed a weak correlation with the confinement parameter adopted. Strains in the tie reinforcement were measured at the side face for some of the medium and high- strength concrete columns. The measured strains were not at yield when the peak load was reached.
This paper explores the load-deformation behaviour of plain and fibrous high-strength reinforced concrete slender columns from zero load until failure. The proposed empirical stress-strain equations given here for high-strength and high-strength steel fibre concretes were used as material properties to modify the computer programs of biaxially loaded slender columns previously developed. The new computer program can evaluate the complete biaxial load-deflection and moment-curvature relationships of slender columns. A total of nine high-strength and five high-strength steel fibre reinforced concrete columns were tested to compare their experimental load-deformation results with the analytical values derived from theoretical studies. Agreement was satisfactory for both ascending and descending branches of the load-deformation curves.
Design Column Using Eurocode 2 Worked Examples
Source: https://www.researchgate.net/publication/250072658_Design_method_for_slender_columns_subjected_to_biaxial_bending_based_on_second-order_eccentricity
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