Circular RC Section Design - N-M Interaction Theory (EC2)
The theory behind this circular reinforced-concrete section calculator: how a round column, pile or beam is checked to Eurocode 2 (EN 1992-1-1) by strain compatibility, how the N-M interaction diagram is built by integrating the circular compression zone, and how slenderness, shear and crack width are verified.
A circular reinforced-concrete section - a round column, a bored pile or a circular beam - is checked to Eurocode 2 (EN 1992-1-1) for the combination of axial force N_Ed and bending moment M_Ed at the ultimate limit state. Because the section is rotationally symmetric and the longitudinal bars sit on a ring, its capacity is captured by a single N-M interaction diagram. This page explains the assumptions, the section geometry, how the capacity is computed by strain compatibility, and the shear and second-order checks behind the calculator.
Assumptions (EN 1992-1-1 §6.1)
The ULS bending-and-axial analysis uses the standard Eurocode 2 assumptions: plane sections remain plane (a linear strain profile); perfect bond between concrete and steel; concrete carries no tension; the concrete stress-strain relation is the parabola-rectangle law of §3.1.7; and the reinforcement is elastic-perfectly-plastic (§3.2.7). Failure is reached when the extreme concrete compression fibre reaches ε_cu2 (3.5‰ for concrete up to C50/60) or the steel reaches its strain limit.
Figure 1 - Strain compatibility: a plane strain profile, the parabola-rectangle concrete stress block over the circular compression segment, and bar forces from their strains.
Section geometry
For a circular section of diameter D and radius R = D/2, the gross concrete area and second moment of area are:
Ac=πR2I=4πR4i=I/Ac=2R
The n longitudinal bars are equally spaced on one ring at a radius
rs=R−c−ϕlink−21ϕ
where c is the nominal cover, φ_link the link (or spiral) diameter and φ the bar diameter. A representative effective depth for a full ring is d = R + (2/π)·r_s, the distance from the extreme compression fibre to the centroid of the tension-side bars.
The compression zone - chord integration
Unlike a rectangle, the width of a circular section varies with depth. At a depth y below the top fibre (with the centre at y = R) the chord width is
w(y)=2R2−(y−R)2
The concrete contribution to the internal axial force and moment is found by slicing the compression zone into thin horizontal strips, each of width w(y), evaluating the strain from the linear profile and the stress from the parabola-rectangle law σ_c(ε), and summing:
Nc=−∫0xσc(ε(y))w(y)dyMc=∫0xσcw(y)(y−R)dy
where x is the neutral-axis depth. Each bar adds a force F_s,i = σ_s(ε_i)·A_s,i at its own depth. This numerical fibre integration is exact for any neutral-axis position and is what the calculator performs (240 strips).
N-M interaction and the moment resistance
To find the moment resistance at a given axial load, the neutral-axis depth is varied until the internal axial force equals N_Ed; the corresponding internal moment is M_Rd. Sweeping the strain profile from full compression to full tension traces the whole N-M interaction envelope. Its anchor points are the squash load and the pure-tension capacity:
NRd,c=fcdAc+fydAsNRd,t=fydAs
The section is adequate when the design point (M_Ed, N_Ed) lies inside the envelope. The moment capacity is largest near the balance point, where the concrete reaches ε_cu2 just as the tension steel yields - a moderate axial compression therefore increases the moment capacity before it falls away towards the squash load.
Figure 2 - N-M interaction envelope. The section is adequate when the design point (M_Ed, N_Ed) lies inside the curve. Moment capacity peaks near the balance point.
Slenderness and second-order effects (§5.8)
A circular column is slender when its slenderness ratio exceeds the limit of §5.8.3.1:
λ=il0=R2l0λlim=n20ABC,n=Acfcd∣NEd∣
If λ > λ_lim, the nominal-curvature method (§5.8.8) adds a second-order eccentricity that magnifies the design moment:
A minimum first-order eccentricity e_0 = max(D/30, 20 mm) (§6.1(4)) is always applied, and geometric imperfections may add e_a = l_0/400.
Shear (§6.2)
EN 1992-1-1 §6.2 is written for rectangular webs, so a circular section is idealised as an equivalent rectangle. A common, code-accepted choice is b_w = D and d = R + (2/π)·r_s, with only the tension-side bars counted in the longitudinal ratio ρ_l. The member-without-links resistance, the link resistance and the strut limit then follow the standard expressions:
with the strut angle θ kept within 21.8° ≤ θ ≤ 45° (i.e. 1 ≤ cot θ ≤ 2.5). Circular links or a continuous spiral provide the shear reinforcement.
Serviceability - stress and crack width (§7.2, §7.3)
At the serviceability limit state the section is analysed cracked and elastic: the compression zone is a circular segment, and the transformed section (concrete plus α_e = E_s/E_cm times the steel) gives the elastic neutral axis. The calculator then checks the concrete stress σ_c ≤ 0.6 f_ck (§7.2), the minimum reinforcement (§7.3.2) and the crack width w_k = s_r,max(ε_sm − ε_cm) against the limit (§7.3.4).
Detailing notes
EN 1992-1-1 §9.5.2 requires a minimum number of longitudinal bars in a circular column (commonly six, with the National Annex governing) and a minimum reinforcement ratio. Bars are tied with circular links or a helical spiral at a pitch limited by §9.5.3. Because the bars sit on a ring, the section resists bending equally about every axis - the practical advantage of round columns under biaxial or wind-governed loading.
Frequently asked questions
A circular RC column is checked at the ultimate limit state for the combination of axial force N_Ed and bending moment M_Ed using an N-M interaction diagram. The section capacity is found by strain compatibility: a plane strain profile is assumed (plane sections remain plane), the parabola-rectangle concrete stress block is integrated over the circular compression zone, and the steel bars on the ring develop forces from their strains. The neutral-axis depth is varied until the internal axial force equals N_Ed, giving the moment resistance M_Rd at that axial load. The design is adequate when the (M_Ed, N_Ed) point lies inside the N-M envelope. This tool builds the full envelope and overlays your design point automatically.
An N-M (axial-moment) interaction diagram is the envelope of all axial-force and bending-moment combinations a section can resist at the ultimate limit state. The vertical axis is axial force (compression and tension) and the horizontal axis is bending moment. The curve is generated by sweeping the strain profile from pure compression (squash load) through the balance point to pure tension. A column is adequate when its design point (M_Ed, N_Ed) plots inside the curve. The bulge near the balance point shows that a moderate axial compression actually increases the moment capacity, before it falls away again towards the squash load.
Unlike a rectangle, the width of a circular section varies with depth: at a depth y from the top fibre the chord width is 2·sqrt(R² − (y − R)²), where R is the radius. To compute the concrete force, the section is sliced into many thin horizontal strips; each strip's width is the local chord, its strain comes from the linear strain profile, and its stress from the parabola-rectangle law (EN 1992-1-1 §3.1.7). Summing the strip forces and their moments about the centroid gives the concrete contribution to N and M. This numerical fibre integration handles any neutral-axis position exactly.
Longitudinal bars in a circular column are placed equally spaced on a single ring (a circle) at a radius r_s = R − cover − link diameter − half the bar diameter. EN 1992-1-1 requires a minimum of bars for a circular column (typically six, with the National Annex governing), and the bars are tied with circular links or a continuous helical spiral. Because the bars sit on a ring, the section behaves identically about every axis - a key advantage of round columns under biaxial bending.
EN 1992-1-1 §6.2 is written for rectangular webs, so a circular section is idealised as an equivalent rectangle. A common, code-accepted approach takes the web width as the diameter (b_w = D) and the effective depth as d = R + (2/π)·r_s, with only the tension-side bars counted in the longitudinal reinforcement ratio ρ_l. The members-without-shear-reinforcement resistance V_Rd,c, the link resistance V_Rd,s and the strut limit V_Rd,max then follow the standard §6.2 expressions. This tool reports all three and the governing value.
The squash load (N_Rd,c) is the pure axial compression capacity with no moment: N_Rd,c = f_cd·A_c + f_yd·A_s, where A_c is the concrete area (πR²) and A_s is the total bar area. It is the lowest point of the N-M interaction diagram. The pure tension capacity is N_Rd,t = f_yd·A_s. Real columns are never designed at the squash load because some moment (from eccentricity or frame action) is always present, but it anchors the top of the interaction curve.
A column is slender, and needs a second-order check, when its slenderness λ = l_0/i exceeds the limit λ_lim from EN 1992-1-1 §5.8.3.1. For a circular section the radius of gyration is simply i = R/2. If λ > λ_lim, the additional second-order moment is found by the nominal-curvature method (§5.8.8): an extra eccentricity e_2 = (1/r)·l_0²/10 is added, magnifying the design moment. This tool computes λ, λ_lim and, when you enable second-order effects, the magnified design moment M_Ed,tot.
Ready to check your own section? Get the N-M interaction diagram, moment capacity, shear and crack width for any circular column or beam.
