Interaction diagrams for reinforced concrete circular cross-section

The procedure for creating interaction diagrams for reinforced-concrete circular crosssections, based on HRN EN 1992-1-1, is described in the paper. Due to introduction of new concrete strength classes, with somewhat different parameters for the stressrelative strain diagram, it has become necessary to develop new interaction diagrams. The computation procedure based on which circular cross section interaction diagrams were made for all concrete classes, was conducted. The use of these interaction diagrams will simplify dimensioning of reinforced-concrete circular cross-sections.


Introduction
New concrete strength classes, with the exception of concrete classes C12/15 to C50/60, are presented in HRN EN 1992 [1], and so new higher concrete classes are introduced: C55/67, C60/75, C70/85, C80/95 and C90/105.This means that, in addition to normal concrete strength classes, this standard takes into account a wider application of higher strength concrete classes, i.e. concrete classes with characteristic compressive cylinder strength of concrete [1] in excess of 50 MPa.
The high strength concrete is less ductile than the normal strength concrete and, consequently, the form of the stress -strain diagram becomes different for higher class concrete grades.In addition, the definition of the design compressive strength of concrete is also different in HRN EN 1992 [1], when compared to HRN ENV 1992 [2].
Because of introduction of new concrete classes with a different stress -strain diagrams for higher class concrete types, it has become necessary to develop new interaction diagrams to enable design of reinforced-concrete circular cross sections with respect to the axial force and bending action.For the same reason, interaction diagrams for design of rectangular cross sections must also be developed.According to information available to the authors, an university course material with interaction diagrams for rectangular sections is soon to be published in Croatia by other authors.As to diagrams presently available in national literature in Croatia [3], we currently have interaction diagrams for circular cross sections according to Jevtić, developed in accordance with the Byelaw for Concrete and Reinforced Concrete [4].
Interaction diagrams for circular cross sections, based on HRN EN 1992-1-1, are already available in literature [5,6].Interaction diagrams for circular cross sections, based on DIN 1045-1 [7], are presented in book [6].Design stressstrain diagrams for concrete, based on standards DIN 1045-1 and HRN EN 1992-1-1, are identical for concrete classes C12/15 to C50/60.Interaction diagrams for circular cross sections according to DIN 1045-1 can be used in Croatia as approximations only, as they are defined as stress -strain diagrams for reinforcing steel with an inclined upper branch, according to DIN 1045-1.Interaction diagrams available in literature [6] are conceived for the biggest relative deformation of tensile reinforcement (0.025 or 25‰), while in Croatia the value of 0.02 (20‰) is normally used.
For concrete classes higher than C50/60 the experimental diagrams for concrete and steel, based on DIN 1045-1 and HRN EN 1992-1-1 are not identical, which is why the interaction diagrams according to DIN 1045-1 can not be used in Croatia.
Interaction diagrams for circular cross sections, based on EN 1992, are given n literature [5].These diagrams are given for concrete classes C12/15 to C50/60, and are related to the zone of eccentric compressive load only.
Interaction diagrams for circular cross sections, based on HRN EN 1992-1-1, are presented in the paper for all classes of concrete, for both eccentric compressive load and tensile load situations.Interaction diagrams were generated using the computer program Mathcad 2001i [8].

Design diagrams for concrete and reinforcing steel 2.1. Design diagram for concrete
According to HRN EN 1992-1-1, the following expression is used to define the design compressive strength: where: f ck -is the characteristic compressive strength of concrete, γ C -is the partial safety factor for concrete, α cc -is the coefficient which takes into account long-term effects on compressive strength and unfavourable effects due to the way in which load is applied.The recommended value according to EN 1992-1-1, as accepted in Croatia' s draft national annex, is α cc = 1,0.
Note: Compressive stress and compressive strain values presented in this paper assume a negative sign, while tensile stress and tensile strain values assume a positive sign.
The stress -strain diagram for concrete, shown in Figure 1, is used for design cross sections with respect to axial force and bending.Compressive strain and compressive stress values for concrete are presented in Figure 1 as absolute values, according to HRN EN 1992-1-1 [1].Concrete properties shown in Table 1 are also presented as absolute values [1].
The stress in concrete can be presented using the following expressions: (1) (2) (3) (4) for for for where n is the exponent, ε c2 is the strain, when the concrete strength has been achieved, while ε cu2 is the ultimate strain value (Table 1).
Interaction diagrams for reinforced concrete circular cross-section

Design diagram for reinforcing steel
Stress -strain diagrams for reinforcing steel, which assume, according to HRN EN 1992-1-1, the same values in tension and compression, are presented in Figure 2.
According to HRN EN 1992-1-1, one of the following two assumptions (Figure 2) can be used in cross-section design: a) inclined upper branch (in compression it is the lower branch) with limited strain ε ud , b) horizontal upper branch without the need to limit strain.
The experimental stress -strain diagram for concrete, with the horizontal top branch and with maximum strain εud = 0.02 (20 ‰), was used as basis for developing interaction diagrams that are presented in this paper.The stress in reinforcing steel (according to Figure 2), for bilinear experimental diagram of steel with horizontal top branch (lower branch in case of compression), amounts to: (5) for for for where the elastic modulus for steel is E s = 200 000 MPa, and f yd is the design yield strength of reinforcing steel.

Interaction diagrams for circular cross sections
The description of the circular cross section with reinforcement is presented in Figs. 3 and 5.The concrete strain in a given point is defined as: The cross-section curvature is defined using the following expression: where: -strain of reinforcement at the bottom edge in Figure 3, -strain of concrete at the top edge in Figure 3, -strain of concrete at the bottom edge in Figure 3, -distance between the centre of gravity of concrete section and the point under study, -total height of cross section (diameter), -distance between the top edge of concrete and reinforcement at the bottom edge, -distance between the uniformly distributed reinforcement and the section edge.
As it is assumed in calculation that plane cross sections stay plane even after deformation (Bernoulli hypothesis of plane sections), all points of the deformed section will be situated in the same plane (plane of strain) which passes through the neutral  In Figure 2, f yk is the characteristic yield strength of reinforcing steel, f yd jis the design strength of reinforcing steel, (f t /f y ) k is the characteristic yield to tensile strength ratio, γ S is the partial factor for reinforcing steel, and ε uk is the characteristic strain of reinforcing steel at maximum load.Figure 2 is valid for tensile and compression stresses and strains.
The differential area of concrete is obtained according to Figure 5: The differential area of reinforcement is defined according to Figure 5, under assumption that the reinforcement is uniformly distributed at the distance rs from the centre of gravity of the concrete section: The design compressive force in concrete is defined according to the following expresssion: The design force in reinforcement is calculated using the following expression: where: A s -total area of reinforcement cross section, r -radius of circular cross section.axis of cross section and is inclined at κ with respect to the undeformed plane of cross section.The strain of reinforcement in a given point of a circle where a longitudinal reinforcement is uniformly distributed is defined according to the following expression (Figs. 3 and 5).
where r s and α are defined in Figure 5. ε c,ed2 , ε s1 or κ , can be defined if the other two values are known.
To define the interaction diagram, two of these values (ε c,ed2 , ε s1 ) must be assumed, i.e. one has to vary the position of the deformation plane which defines strains and stresses in concrete and reinforcement.
Interaction diagrams can be used to design of cross sections subjected to compressive or tensional axial forces with bending moment.They can also be used to design of cross sections subjected to bending action without axial force, and cross sections subjected to axial force only.This dimensioning can be made in various positions of deformation plane.Zones of deformation plane for cross section of general shape are shown in Figure 4. Strain plane positions were varied by rotating the strain plane around axes parallel to neutral axis of cross section, and they pass through points A, B or C (Figure 4): Zone 1 (rotation around point A) Interaction diagrams for reinforced concrete circular cross-section where: -design acting axial force, -design acting bending moment, -resistance of cross section to axial force action, -bearing moment of cross section.
The resistance of cross section to axial force action is: . When expression ( 17) is inserted into the expression (15), we obtain: , i.e. when expressions ( 13) and ( 14) are inserted for F cd and F sd into the previous expression (18), we obtain: When the expression ( 19) is divided by the factor A c × f cd , we obtain: where A c = h 2 π/4 area of the concrete cross section.
The dimensionless axial force ν Ed and the mechanical reinforcement ratio ɷ are introduced into the expression (20): and so for N Ed = N Rd we obtain: The load bearing moment by which the cross section withstands the bending action amounts to: i.e.: Once the expression ( 26) is inserted into the expression ( 16) we obtain: then the expression ( 27) is divided by the factor A c × h× f cd : The mechanical reinforcement ratio ɷ (expression ( 22)) and the dimensionless bending moment μ Ed are inserted into the expression (28): and so the algebraic connection between the dimensionless bending moment μ Ed and the mechanical reinforcement ratio ɷ is obtained for M Ed = M Rd : The required cross section area of reinforcing bars, uniformly distributed at the distance r s from the centre of gravity of concrete, is determined using the following expression: The interaction diagrams shown in Figures 6 to 11 were calculated and developed using the equilibrium equations ( 24) and (30).The steel ε s1 and concrete ε c,ed2 deformation (curvature κ) and the mechanical reinforcement ratio ɷ were varied, and the values ν Ed and μ Ed were calculated and inserted into the coordinate system ν Edμ Ed .Interaction diagrams were developed for the circular cross section and concrete classes C12/15 to C50/60, C55/67, C60/75, C70/85, C80/95 and C90/105, for the steel B500, and for the ratio of d 1 /h = 0,1.Parameters d 1 and h are defined by the expression (9).
The equilibrium equations ( 24) and (30) were set up in such a way that they apply for all three cross section deformation planes (Figure 4).Analytical solutions for integrals contained in equations ( 24) and (30) were not derived as they would be too extensive considering the circular shape of cross section, and so interaction diagrams were calculated using the MathCad software through which numerical integrations were made.
The concrete stress values σ c , as given in expressions ( 24) and (30), were integrated along the entire height of the cross section, i.e. from -h/2 to h/2.The reinforcement stress values σ s as given in expressions ( 24) and (30), were integrated along the entire circle, i.e. from the angle α = 0 to 2π.Stress values for concrete were determined using expressions (2) to (4), while stresses in steel were defined according to expressions (5) to ( 7).

Numerical example
The circular cross section of the column 50 cm in diameter, with reinforcement uniformly distributed at the distance d1 = 5 cm from the edge of the cross section, has to be dimensioned.The column will be made of concrete class C30/37 and reinforcing steel B500B.The cross section under study is subjected to the following design section forces: M Ed = 392 kNm i N Ed = -1570 kN.

Ratio:
The design compressive strength of concrete:

Figure 5 .
Figure 5. Differential areas of concrete reinforcementTwo conditions must concurrently be met in sections that are simultaneously subjected to bending moment and axial force:(15)(16)

Figure 3 .
Figure 3. Circular cross section with distribution of strains, stresses and forces As shown in expression (9) any one of the three values, i.e.

Figure 4 .
Figure 4. Possible strain distributions in reinforced-concrete of general shape according to (2)