The use of annular and circular cross-section members in transport and power engineering building construction/Ziedinio ir skritulinio skerspjuvio konstrukciju panaudojimas transporto ir energetikos statyboje/Gredzenveida un apala skersgriezuma elementi transporta un elektroenergetikas buvem/Ronga-ja ringikujulise ristloikega elementide kasutamine transpordi ja energeetikaga seotud ehitistes.
Vadluga, Romualdas ; Kliukas, Romualdas
1. Introduction
Solid circular columns are very popular for bridge pier design
because the construction is simple and its strength characteristics
under wind and seismic loads are similar in any direction. Furthermore,
circular elements are also widely used as columns in buildings or as
piles for foundations. Hollow core circular concrete members are much
less structurally used than solid circular cross sections. Though, these
can be found in concrete chimneys, concrete pipes and elevated water
tanks, as well as in large bridge columns and offshore platforms (Turmo
et al. 2009).
Hollow core circular concrete elements are economically and
conveniently manufactured using spin casting technique. Spin casting is
an effective method to produce concrete pylons, masts or pipes. Through
the centrifugation process the concrete is compacted and the desired
shape, mostly round or ellipsoidal, is obtained (Abeles 1973; Bacsa
1979; Bacsa 1981; Kaufmann, Hesselbarth 2007; Kvedaras, Sapalas 1999;
Wesley, Wong 2002).
Normally, the load-bearing reinforced concrete structures of
buildings and other engineering constructions are of a circular (round)
shape (Fig. 1). The use of structural elements of this shape in building
construction in the areas of transport and power engineering has some
advantages. Since the buildings intended for use in transport and power
engineering are constructed in the open areas, their structures are
exposed to weather (Dilger et al. 1996; Selih 2010). The perimeter of a
circular cross-section is by 12.8% smaller than the perimeter of the
rectangular cross-section of the same area. Therefore, the surface area
of circular cross-section elements exposed to weather is smaller than
the surface area of the rectangular (square) cross-section elements.
Another factor ensuring higher durability of the considered
structures is that they have a smooth surface without any ribs, typical
of round elements. Field tests of long-service power transmission lines
aimed at determining the technical state of electric power pylons have
shown that the characteristic cross cracks were formed in the corners of
the rectangular cross-sections near the longitudinal reinforcing bars
(Kliukas et al. 2003; Vadluga, Kliukas 2000). The tests have shown that
the technical state of the spun concrete pylons of annular cross-section
was much better than the state of concrete members of the rectangular
cross-section made of concrete compacted by vibration.
As shown by numerous experimental and theoretical studies
(Kvedaras, Kudzys 2006; McAteer et al. 2004; Mei et al. 2001;
Pellegrino, Modena 2010; Selih 2010; Wang, Wu 2008), higher durability
of structures is achieved by applying an external cover, typically of
circular shape, which may be made of metal, plastic or reinforced
concrete.
Concrete generally has to be reinforced with steel bars which are
susceptible to corrosion. Furthermore, the placement of the steel
reinforcement is time-consuming and hence expensive. The structural
elements of this type are rather thick and heavy. The application of
short-fiber reinforced cement of carbon or polyvinyl alcohol (Domagata
2011; Kaufmann et al. 2005) is a suitable alternative for such slightly
bended elements.
[FIGURE 1 OMITTED]
Codes do not provide calculation methods suitable in all cases to
evaluate the strength (shear strength) of circular cross-section
elements of reinforced concrete (AC/ 318-05:2005 Building Code
Requirements for Structural Concrete; BS8 100:1985 Structural Use of
Concrete: Part I: Code for Practice for Design and Construction; EHE
1998:1998 EHE Instruction for Structural Concrete; CEB-FIP 1990:1993
CEB-FIP Model Code; SNiP 2.05.03-84:1985 Building Code. Bridges and
Pipes; Rodgers et al. 1983). The calculation of the reinforced concrete
members of circular cross-section is more complicated than the
calculation of the rectangular cross-section elements. The circular
shape of the cross-section and the uniform distribution of the
reinforcing bars along the perimeter impart some specific features to
the evaluation of the stress-strain state of the members (Harajli 2006;
Mander et al. 1988; Pantazopoulou, Mills 1995). Since the longitudinal
bars are uniformly distributed along the cross-section perimeter, the
variation of their stress-strain state is accompanied not only by the
variation of the parameters of the compressed zones, but the ratio of
the areas of the longitudinal bars found in the compression and tension
zones as well.
[FIGURE 2 OMITTED]
Reinforced concrete elements of circular (solid) cross-section and
the elements of annular (hollow) cross-section are usually compressed.
Therefore, both their strength and stability, depending on flexural
stiffness (EI) are important. In this respect, the second moment of area
(I) of the solid circular cross-section elements is by about 4.5%
smaller than that of the elements of the rectangular cross-section
having the same area (Arslan, Cihanli 2010; Kliukas et al. 2010; Kudzys,
Kliukas 2010; Kvedaras, Kudzys 2010). This difference may be smoothed by
using annular cross-section elements, whose ratio of the second moment
of area and cross-section area is higher, when the ratio of their
internal and external diameters is higher (Fig. 2).
A feasibility study of using annular cross-section reinforced
concrete elements in building construction in transport was described in
two articles published earlier in the Baltic Journal of Road and Bridge
Engineering (Kudzys, Kliukas 2008a; 2008b). In the present paper, a
relatively simple method of calculating reinforced concrete members of
circular cross-section, based on the main principles used in calculating
annular cross-section elements, is described.
2. The main assumptions of the method of calculation
In calculating the bearing capacity of reinforced concrete
structures by the limiting state method, it is assumed that the element
is at the stage of failure, but concrete behaviour under tension is not
taken into account. For the sake of calculation simplicity, the diagram
presenting concrete stresses in the compressed zone of the element by
curves is replaced with a diagram of arbitrary rectangles. The same
principle is also used in the EN 1922-1:2004 Eurocode 2. Design of
Concrete Structures--Part 1: General Rules and Rules for Buildings.
Unlike the rectangular cross-section elements, whose longitudinal
reinforcing bars are located as close as possible to the most highly
stressed layers of the compression and tension zones, the longitudinal
reinforcing bars of the circular cross-section elements are usually
uniformly distributed along the circular cross-section perimeter. The
diagrams of the reinforcement stresses in the compression and tension
zones of such elements at the stage of failure are curve diagrams, i.e.
the strength of the reinforcement is used depending on its position in
the cross-section. The bearing capacity of an element may be evaluated
by a universal method of strength calculation, presented in the
Lithuanian construction specifications STR 2.05.05:2005 Design of
Concrete and Reinforced Concrete Structures, based on the previously
valid structural design codes and specifications CHu[PI] 2.03.01-84.
However, the calculation based on the considered method is complicated
and rarely used.
3. Bayes theorem in revised reliability analysis
In actual calculations, the curve diagrams of concrete and
reinforcement stresses (Fig. 3) are replaced with the diagram of
arbitrary rectangles (Vadluga 2007). The compressed cross-section zone
is defined by its sector portion. In this case, the position of the
neutral axis, i.e. the relative value [xi] (Fig. 2) is hard to
determine. The problem is solved by using the successive approximation
method. The solution is complicated because the relative value
[[xi].sub.s] describing the compressed part of the reinforcement of the
element cross-section does not match the value [xi].
The relationship between these values is expressed by the formula:
[[xi].sub.s] = arccos([r/[r.sub.s]]cos[pi][xi])/[pi] = k[xi] (1)
The values of the coefficient k, characterizing the relationship
between [[xi].sub.s] and [xi], depend on the ratio r/[r.sub.s]. They are
given in Table 1.
[FIGURE 3 OMITTED]
As shown by the data given in Table 1, the difference between
[[xi].sub.s] and [xi] is the greatest, when the ratio r/[r.sub.s]
increases. When the relative size of the compressed zone is in the range
of [xi] = 0.3-0.7, the value of the coefficient k does not differ from
unity by more than 10% even for the worst case (when r/[r.sub.s] =
1.20). In the case of concrete-filled members with tubular reinforcement
(Saraskinas, Kvedaras 2000), the values of [[xi].sub.s] and [xi] match
each other.
The value of [xi], characterizing the compressed cross-section area
of the member, is determined from the equilibrium condition, stating
that the sum of all axial (internal and external) forces is equal to
zero:
[f.sub.cd][A.sub.c] + [f.sub.sc,d][A.sub.sc] - [f.sub.yd][A.sub.st]
- N = 0, (2)
where [f.sub.cd]--the calculated compression strength of the
concrete, [f.sub.sc,d] and [f.sub.yd] denote the calculated compressive
and tensile strength of the reinforcement, respectively, kN/[m.sup.2];
[A.sub.c]--the compressed zone of the elements cross-section area,
[m.sup.2]; [A.sub.sc] and [A.sub.st]--the cross-section areas of the
longitudinal compression or tension reinforcing bars respectively,
[m.sup.2]. They may be expressed as follows: [A.sub.sc] =
[A.sub.s][[xi].sub.s] and [A.sub.st] = [A.sub.s](1 - [[xi].sub.s]),
where [A.sub.s]--the total cross-section area of the element's
reinforcement, [m.sup.2]; N--normal axial compressive force, kN.
[FIGURE 4 OMITTED]
The area of the compressed zone of the element's cross-section
is described as the area of a segment of a circle:
[A.sub.c] = [[r.sup.2]/2](2[pi][xi] - sin2[pi][xi]) =
[pi][r.sup.2]([xi] - sin[pi][xi]cos[xi]/[pi]) (3)
where [pi][r.sup.2]--the total area of the element's
cross-section.
By substituting the expressions for the above-mentioned
characteristics into the Eq (2), the equation allowing us to calculate
the value [xi] characterizing the compressed part of an element's
cross-section is obtained. The solution can only be made by using the
successive approximation method. For this purpose, a computer program
based on the algorithm presented in Fig. 4 can be made.
For practical calculations, a simplified approximate method can be
used. The analysis has shown that, in the interval [xi] = 0.3-0.7, the
expression between the brackets of the Eq (3) may be replaced with a
simpler expression rather accurately (Vadluga 2007).
[xi] - [sin[pi][xi]cos[pi][xi]/[pi]] [approximately equal to]
1.8[xi] - 0.4 (4)
Then, the equilibrium condition (2) may be rewritten as
[f.sub.cd]A(1.8[xi] - 0.4) + [f.sub.sc,d][A.sub.s][[xi].sub.s] -
[[xi].sub.yd][A.sub.s](1 - [[xi].sub.s]) - N = 0 (5)
or
[f.sub.cd]A(1.8[xi] - 0.4) + [f.sub.sc,d][A.sub.s]k[xi] -
[f.sub.yd][A.sub.s] (l - k[xi]) - N = 0. (5a)
After solving the Eq (5a), the relative size of the compression
zone of the element may be defined as follows:
[xi] = [0.4[f.sub.cd]A + [f.sub.yd][A.sub.s] + N]/[l.8[f.sub.cd]A +
k([f.sub.sc,d] + [f.sub.yd]) [A.sub.s]]. (6)
The bearing capacity of the reinforced concrete members of circular
cross-section is obtained based on the equilibrium condition. This means
that the sum of the moments, which develop due to the action of the
internal and external forces about the axis, passing through the centre
of the member's cross-section, is equal to zero:
[M.sub.0] = [([Ne.sub.0]).sub.u] = [2/3]
([f.sub.cd][r.sup.3][sin.sup.3][pi][xi]) + ([f.sub.yd] + [f.sub.sc,d])
[A.sub.s][r.sub.s] [sin[pi][[xi].sub.s]/[pi]] (7)
The suggested method has been checked by comparing the bearing
capacity of eccentrically compressed reinforced concrete members of
circular cross-section, calculated using this method, with the
experimental data presented in the literature on the problem (Sapalas
1978). It has been found that the suggested method is more suitable when
the eccentricity [e.sub.0]/r is smaller than 0.3. The comparative
analysis has shown that the ratio of the calculated bearing capacity
values of eccentrically compressed concrete elements of circular
cross-section, obtained by using the suggested method, and the
respective experimental data is as follows: the mean ratio value is m =
1.0686, and the mean square deviation is [bar.s] = 0.0709. The
confidence interval of this ratio according to the Student t-test is
0.9918-1.1454 (Vadluga 2007).
Further analysis of this problem has shown that the Eq (7) is not
universal. This particularly refers to the calculation of the bearing
capacity of the members eccentrically compressed with small
eccentricity.
A general (universal) formula is obtained by considering the
equilibrium of the moments about the axis, passing through the centre of
the reinforcement zone subject to tension or slight compression. In
calculating the bearing capacity of the concrete elements (without the
longitudinal reinforcing bars), the equilibrium condition about the axis
passing through the conventional centre of the concrete element's
zone in tension is considered, i.e. it is assumed that [r.sub.s] = r. It
follows from the equilibrium condition that the sum (equal to zero) of
the moments, which develop due to the action of the external and
internal forces about the axis, passing through the element's
cross-section reinforcement centre subject to tension or slight
compression, is obtained as follows:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)
The suggested method was checked by comparing the calculated
bearing capacity of eccentrically compressed plain and reinforced
concrete members of circular cross-section with the experimental data
(Sapalas 1978, Docenko 1954). The results of the comparative analysis
are presented in Tables 2 and 3.
The analysis performed has shown that the method of calculating the
bearing capacity of the considered members by the Eq (8) is more
accurate and reliable. Based on the suggested method, the bending,
compression and tensile strength of reinforced concrete and concrete
members of circular cross-section may be calculated irrespective of the
eccentricity value.
4. The comparison of methods for calculating the bearing capacity
of circular and annular cross-section members
In calculating the bearing capacity of the reinforced concrete
elements of annular cross-section, a section of the compressed zone is
described as the part of a sector rather than a ring section. This is
done for the sake of margin (Kudzys 1975). The latter assumption does
not suit when calculating the bearing capacity of circular cross-section
elements. However, when the neutral axis of circular cross-section
members does not cross the internal circumference of the ring (implying
that the height of the compressed zone is smaller than the thickness of
the ring (Fig. 3)) and the bearing capacity of annular cross-section
members may be calculated similarly to the calculation of circular
cross-section members.
5. Conclusions
A relatively simple method of calculating the bearing capacity of
circular cross-section reinforced concrete elements is suggested. It has
been shown that this method is analogical to the method of calculating
the bearing capacity of annular cross-section reinforced concrete
elements. The suggested method may be used for calculating circular
cross-section reinforced concrete elements in testing and design.
DOI: 10.3846/bjrbe.2012.11
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Received 20 April 2011; accepted 25 May 2011
Romualdas Vadluga (1), Romualdas Kliukas (2) ([mail])
(1) Dept of Reinforced Concrete and Masonry Structures, Vilnius
Gediminas Technical University, Sauletekio al. 11, 10223 Vilnius,
Lithuania
(2) Dept of Strength of Materials, Vilnius Gediminas Technical
University, Sauletekio al. 11, 10223 Vilnius, Lithuania
E-mails: (1) gelz@vgtu.lt; (2) romualdas.kliukas@vgtu.lt
Table 1. The values of the coefficient k describing
the relationship between [[xi].sub.s] and [xi]
[xi]\k k, when r/[r.sub.s]
1.05 1.1 1.15 1.2 1.25
0.1 0.1679 - - - -
0.15 0.7660 0.6360 - - -
0.2 0.8846 0,7538 0.5974 0.3854 -
0.25 0.9346 0,8653 0.7910 0.7100 0.6197
0.3 0.9609 0.9207 0.8791 0.8360 0.7910
0.35 0.9767 0.9530 0.9290 0.9046 0.8798
0.4 0.9870 0.9740 0.9609 0.9477 0.9344
0.45 0.9944 0.9889 0.9832 0.97775 0.9719
0.5 1.0000 1.0000 1.0000 1.0000 1.0000
0.55 1.0046 1.0092 1.0138 1.0184 1.0230
0.6 1.0086 1.0173 1.0261 1.0349 1.0437
0.65 1.0126 1.0253 1.0382 1.0514 1.0647
0.7 1.0167 1.0340 1.0518 1.0703 1.0896
0.75 1.0218 1.0449 1.0697 1.0967 1.1268
0.8 1.0288 1.0615 1.1006 1.1536 -
0.85 1.0413 1.017 - - -
0.9 1.0925 - - - -
Table 2. The results obtained in determining the ratio of the
calculated and experimental bearing capacity of the elements
Mean value
Method Element Quantity [[rho].sub.m] = Mean square
([M.sub.obs]/ deviation
[M.sub.cal]) [s.sub.[rho]]
Eq (7) reinforced 24 1.089 0.12
concrete
plain 13 0.958 0.108
concrete
Eq (8) reinforced 24 1.051 0.048
concrete
plain 13 0.979 0.057
concrete
([k.sub.min]) *
Method Variation ae = 2 ae = 3
coefficient
[c.sub.p]
Eq (7) 0.11 1.179 1.373
0.113 1.347 1.576
Eq (8) 0.045 1.046 1.101
0.038 1.104 1.151
* Note. The min confidence coefficient of the calculation method
for the case, when the distribution [rho] is assumed to be normal,
is [k.sub.min] = 1/[[rho].sub.m] - ae [s.sub.rho], where ae is
the number of mean square deviations.
Table 3. The confidence interval of the relative mean value
([[rho].sub.m]) * according to the Student t-test
Members Probability
p = 0.95 p = 0.99 p = 0.998
reinforced concrete 1.031-1.071 1.024-1.078 1.009-1.093
plain concrete 0.957-1.002 0.948-1.010 0.935-1.023
* Note. The calculated bearing capacity is obtained from Eq (8).
