STRENGTH ANALYSIS OF REINFORCED CONCRETE COLUMNS OF
CIRCULAR CROSS-SECTION
R. Vadlūga
Dept of Reinforced Concrete and Masonry Structures, Vilnius Gediminas Technical University,
Saulėtekio al. 11, LT-2040 Vilnius. E-mail: gelz@st.vtu.lt
Received
accepted
Abstract. Assessment of carrying capacity of reinforced concrete columns of circular cross-section is considered in
this article. Strength analysis of reinforced concrete columns of circular cross-section is specific in comparison with
these of rectangular cross-section. The proposed method of analysis is analogous to that of the ring cross-section
columns. Carrying capacity of the circular cross-section columns obtained from calculation by proposed method
agrees well with experimental results.
Keywords. Circular cross-section, plain concrete and reinforced concrete columns, strength, carrying capacity,
longitudinal reinforcement
1. Introduction
Reinforced concrete columns of circular cross-section are
widely used in the buildings of framed structure. Analysis
of such columns is more complicated than that of
rectangular members. Circular shape of the cross-section
and uniform distribution of reinforcement along the
perimeter of the cross-section create some peculiarities in
assessment of state of stresses and deformations. In the
engineering literature the problem of analysis of such
members is not enough investigated [1-5]. In this article
quite simple engineering method for analysis of
reinforced concrete columns of circular cross-section
based on the same principles as in analysis of the ring
cross-section members is presented.
2. Assumptions for the method of analysis
In the carrying capacity analysis of reinforced concrete
structures by the limit state (partial coefficient) method it
is assumed that the member is in the state of failure and
the influence of the concrete in tension is ignored.
Diagram of stress distribution in the compression zone is
curved but for the sake of simplification of analysis the
actual curved diagram is superseded by arbitrary
rectangular one. Such principal is utilized and in the
Eurocode 2 [6].
The longitudinal reinforcement in the members of
circular cross-section usually is uniformly distributed in
the perimeter of the cross-section. In the failure stage of
such members the diagrams of stresses of reinforcement
in the tensile and compressive zones are curved, i.e.
utilization of reinforcement strength depends on its
location in the cross-section. Carrying capacity of the
member can be evaluated using the general method for
strength analysis presented in Lithuanian technical
regulation for construction [7] according to the code that
was valid earlier [8]. But analysis using this method is
quite complicated and seldom applied.
3. Simplification of analysis
In practical analysis the said curved diagrams of concrete
and reinforcement strength are superseded by arbitrary
rectangular ones (Fig. 1). Compression zone in the crosssection is defined by a sector part of the cross-section. In
this case determination of location of the neutral axis, i.e.
the value of ξ (see Fig. 1), is determined in a quite
complicated way. The problem is solved by the iteration
method. The solution is impeded by the fact that the
relative value of ξ s defining the compressive part of
reinforcement in the cross-section does not coincide with
the value of ξ . Relationship between these two values is
expressed by formula:
⎛r
⎞
arccos ⎜ 2 cos πξ ⎟
⎝ rs
⎠ = kξ
ξs =
π
(1)
N
f sc , d
πξ s
e0
f cd
x
πξ
rs
f yd
Fig. 1. Diagram for analysis of eccentrically compressed
members of circular cross-section
Values of coefficient k describing relationship between
factors ξ s and ξ depend on the ratio r2 / rs and are
presented in the table.
ξ−
Table 1. Values of coefficient k describing relationship
between factors ξ s and ξ
k at r2 / rs
k
ξ
1,05
0,1679
0,7660
0,8846
0,9346
0,9609
0,9767
0,9870
0,9944
1,0
1,0046
1,0086
1,0126
1,0167
1,0218
1,0288
1,0413
1,0925
0,1
0,15
0,2
0,25
0,3
0,35
0,4
0,45
0,5
0,55
0,6
0,65
0,7
0,75
0,8
0,85
0,9
1,1
0,6360
0,7538
0,8653
0,9207
0,9530
0,9740
0,9889
1,0
1,0092
1,0173
1,0253
1,0340
1,0449
1,0615
1,017
-
1,15
0,5974
0,7910
0,8791
0,9290
0,9609
0,9832
1,0
1,0138
1,0261
1,0382
1,0518
1,0697
1,1006
-
1,2
0,3854
0,7100
0,8360
0,9046
0,9477
0,97775
1,0
1,0184
1,0349
1,0514
1,0703
1,0967
1,1536
-
1,25
0,6197
0,7910
0,8798
0,9344
0,9719
1,0
1,0230
1,0437
1,0647
1,0896
1,1268
-
Data in the table demonstrate that the difference between
ξ s ir ξ increase with the ratio of r2 / rs . When the value
of relative compression zone is within the limits of
ξ = 0,3 ÷ 0, 7 value of the coefficient k does not differ
from unity more than 10%.
In the case of concrete core members with external
reinforcement in the form of a thin steel pipe [9] the
values of ξ s and ξ coincide.
The value of ξ defining the compressive part of the
cross-section is determined from the condition of
equilibrium that the sum of all (internal and external)
forces is equal to zero:
f cd Ac + f sc , d Asc − f yd Ast − N = 0 .
(2)
Here f cd - concrete design compressive strength, f sc , d
and f yd
-
design
strengths
of
reinforcement
in
compression and tension correspondingly, Ac - zone of
cross-sectional area in compression of the member, Asc
and
Ast - cross-sectional areas of longitudinal
reinforcement
in
compression
and
tension
correspondingly.
Area of zone in compression of cross-section of the
member is determined as the area of a segment:
Ac =
r22
sin πξ cos πξ ⎞
( 2πξ − sin 2πξ ) = πr22 ⎛⎜ ξ −
⎟.
2
π
⎝
⎠
brackets of the formula (3) may be changed by the
simpler one:
(3)
In this formula πr22 means the total cross-sectional area
A of the member. Analysis showed that for values of ξ
within the interval of ξ = 0,3 ÷ 0, 7 expression in the
sin πξ cos πξ
≈ 1,8ξ − 0, 4 .
π
(4)
Then condition of equilibrium (2) can be rewritten as
follows:
f cd (1,8ξ − 0, 4 ) + f sc , d Asc ξ s − f yd Ast (1 − ξ s ) − N = 0 , (5)
or
f cd (1,8ξ − 0, 4 ) + f sc , d Asc k ξ − f yd Ast (1 − k ξ ) − N = 0 . (5a)
Solution of this equation:
ξ=
0, 4 f cd A + f yd As + N
(
)
1,8 f cd A + k f sc , d + f yd As
,
(6)
here A = πr22 - cross-sectional area of the member, As the total cross-sectional area of longitudinal
reinforcement.
Load carrying capacity of circular columns is determined
from condition of equilibrium that the sum of the
moments due to external and internal forces about the
axis through the center of cross-section of the member
equals to zero:
( Ne0 )u =
sin πξ s
2
f cd r23 sin 3 πξ + f yd + f sc , d As rs
.(7)
3
π
(
) (
)
4. Approbation of the method
Analysis revealed that the design value of load carrying
capacity of eccentrically compressed reinforced concrete
members of circular cross-section determined by the
proposed method is very close to that determined using
other methods presented in the technical literature [1-4].
The method was verified by comparison of design load
carrying capacity of eccentrically compressed concrete
members of circular cross-section determined by the
proposed method with experimental data [5]. It was
revealed that the proposed method is more suitable when
the relative eccentricity e0/r is less than 0,3. The
comparison showed that the average ratio between design
load carrying capacities of eccentrically compressed
reinforced concrete members of cylindrical cross-section
determined by the proposed method and experimental
ones m = 1, 0686 , and standard deviation s = 0, 0709 .
Confidence interval of the mean ratio with the guarantee
of 99% according to the Student t-criterion is
0,9918 − 1,1454 .
5. Analogy of methods of analysis for circular and
ring cross-section members
References
1.
In load carrying capacity analysis of ring cross-section
reinforced concrete member the compression zone part is
determined not by a segment of the ring but by part of
the sector. It gives non-significant error for the sake of
safety [10]. This assumption is not acceptable for load
carrying capacity evaluation of members with circular
cross-section. But the neutral axis of ring cross-section
members does not cut the internal circle of the ring, i.e.
when the depth of compression zone is less than the
thickness of the ring (see Fig. 2) and then load carrying
capacity of the ring cross-section members can be
determined as for these of circular cross-section.
πξ s πξ
rs
rs
r1
r2
Fig. 2. Ring cross-section of reinforced concrete members
5. Conclusions
Prepared relatively simple method for load carrying
capacity analysis of reinforced concrete columns of
circular cross-section is the result of performed
investigation. Analogy of proposed method with the
method for strength analysis of such members with the
circular cross-section is demonstrated. Method can be
used for verifications and for direct (design) calculations
of reinforced concrete columns of circular cross-section.
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Technical
Regulation
for
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