Reinforced concrete column — N + My + Mz

Check of an isolated braced reinforced-concrete column of rectangular section with symmetrical reinforcement in compression with biaxial bending to ČSN EN 1992-1-1 including the National Annex: imperfections, slenderness, second-order effects by the nominal curvature method, an N–M interaction diagram and basic detailing rules. Internal forces are entered manually from an external structural analysis.

1 — Materials

Concrete class

Maximum aggregate size dgmm

Reinforcement B500B: fyk = 500 MPa, fyd = 434.8 MPa, Es = 200 GPa. Stress-strain diagram with a horizontal top branch and no strain limit (3.2.7(2)b). Factors γc = 1.5, γs = 1.15 and αcc = 1.0 to the National Annex (NA.2.8, NA.2.12).

Refine the clear bar spacing

Clear spacing factor k1[-]

The minimum clear bar spacing is max(k1·Ø, dg + 5 mm, 20 mm) to 8.2(2); the National Annex sets k1 = 1.2 (NA.2.76).

2 — Cross-section

Section width bmm

Section depth hmm

Nominal cover to the links cnommm

Link diameter Øswmm

Link spacing smm

The y axis runs along the width b, the z axis along the depth h. My bends the section in the plane of the depth h — slenderness λy = l0,y/iy, iy = h/√12. Mz bends it in the plane of the width b — λz = l0,z/iz, iz = b/√12.

3 — Longitudinal reinforcement

Bar diameter on the b faces Ø1mm

Bars per b face incl. corners nbpcs

Diameter of intermediate bars on the h faces Ø2mm

Bars per h face incl. corners nhpcs

The corner bars have the diameter Ø1 and belong to the b faces, so the column has 2·nb + 2·(nh − 2) bars. Every bar touches the link. The diameter Ø2 applies only for nh ≥ 3.

4 — Lengths and imperfections

Column length Lm

Effective length — bending My l0,ym

Effective length — bending Mz l0,zm

Geometric imperfection

Braced columns only: l0 ≤ L. The length L gives the imperfection reduction αh = 2/√L (5.2(5)) and the position of the moment maximum along the member.

5 — ULS internal forces

Axial force, compression > 0 NEdkN

Moment My at the top My,1kNm

Moment My at the bottom My,2kNm

Moment Mz at the top Mz,1kNm

Moment Mz at the bottom Mz,2kNm

NEd > 0 is compression. My > 0 compresses the +z face, Mz > 0 compresses the +y face (axes in the section). Enter both end moments in the same convention as in the bending moment diagram: equal signs mean single curvature, opposite signs double curvature. Convert member-end moments from FEM programs before entering them.

First-order moments vary linearly between the top and the bottom — no transverse load along the member.

6 — Creep

Final creep coefficient φ(∞,t0)[-]

Axial force — quasi-permanent NEqpkN

Moment My at the top — quasi-permanent M0Eqp,y,1kNm

Moment My at the bottom — quasi-permanent M0Eqp,y,2kNm

Moment Mz at the top — quasi-permanent M0Eqp,z,1kNm

Moment Mz at the bottom — quasi-permanent M0Eqp,z,2kNm

Quasi-permanent first-order end moments without imperfection, in the same convention as the ULS moments. The effective creep ratio φef = φ(∞,t0)·M0Eqp/M0Ed is taken at the end where M02 acts, both values including the imperfection (5.8.4(2), (3)).

Column section

N–M interaction diagram

N–My (y axis)N–Mz (z axis)

Assumptions and scope

Isolated braced column (l0 ≤ L) with a constant axial force and a constant section including the reinforcement; global second-order effects are checked elsewhere.
First-order moments vary linearly between the ends; the imperfection is added as an eccentricity ei with both signs and the worse variant governs (5.2, 5.8.1).
Second-order effects by the nominal curvature method: MEd = max(max along the member |M0(x)| + M2(x), M0e + M2, NEd·e0) (5.8.8, 6.1(4)).
Section resistance from a plane strain distribution: parabola-rectangle concrete without tension, reinforcement with a horizontal branch and no strain limit, every bar at its position (6.1, 3.1.7, 3.2.7(2)b).
Biaxial bending to 5.8.9: exemption (5.38a,b), otherwise criterion (5.39) in two imperfection variants.
Out of scope: unbraced columns, the general method and the nominal stiffness method, tension, transverse load along the member, shear, torsion, fire, seismic action, fatigue, serviceability limit states, cover durability, laps, anchorage and cross-ties.

The compressive force NEd is positive; enter the end moments in the same convention as in the bending moment diagram — the same sign means tension on the same side.
The effective lengths l0,y and l0,z are entered by the user; the calculator does not derive them from the end restraints.
The calculator does not check shear, torsion, fire resistance, cover durability, accidental situations, unbraced columns or global second-order effects; it is not a substitute for a complete structural design.