# Designing Reinforced Concrete Compression Elements Subjected to Biaxial Bending with the Nominal Curvature Method

### Technical Article

Daily tasks in reinforced concrete design also include designing compression elements subjected to biaxial bending. The following article describes the different methods according to Chapter 5.8.9, EN 1992-1-1, which can be used to design compression elements with biaxial load eccentricities by means of the nominal curvature method according to 5.8.8.

#### General

The RF-/CONCRETE Columns add-on module designs reinforced concrete compression elements by means of the nominal curvature method as described in Eurocode 2, Chapter 5.8.8. Other standards describe this method as model column method. A previous article describes in detail the determination of load eccentricities which have to be applied when using the nominal curvature method. For this reason, determining the single eccentricities will not be described in detail here.

#### Separate Design in Direction of the Principal Axis with Considering Biaxial Moment Interaction

If the conditions of Equation 5.38a and Equation 5.38b [1] are not fulfilled, the prerequisites for the separate design in the principal axis direction without considering the biaxial moment interaction according to 5.8.9 (2) are not met. Section (4) of Chapter 5.8.9 [1] uses Equation 5.39 to describe a simplified approach which you can use to consider the biaxial moment interaction with previous design of the individual principal axis directions.

The following Equation 5.39 [1] takes the moment interaction into account in a simplified way.

$\left[\frac{{\mathrm M}_{\mathrm{Ed},\mathrm z}}{{\mathrm M}_{\mathrm{Rd},\mathrm z}}\right]^{\mathrm a}\;+\;\left[\frac{{\mathrm M}_{\mathrm{Ed},\mathrm y}}{{\mathrm M}_{\mathrm{Rd},\mathrm y}}\right]^{\mathrm a}\;\leq\;1$
where
MEd,z/y = design moment about the corresponding axis, including nominal second order moments
MRd,z/y = bending resistance about the corresponding axis
a = 2 for circular and elliptical cross-sections, according to Figure 02 for rectangular cross-sections

Figure 02 shows the exponent a as a function of the ratio NEd/NRd. NRd is the design value of the centric axial resistance and can be determined with NRd = Ac ⋅ fcd + As + fyd. Ac represents the gross cross-section area, As the longitudinal reinforcement area, and fcd and f yd the design strengths of the used materials.

When using Equation 5.39 [1], please note that the two bending resistances MRdy and MRdz have to be taken from the design interaction diagrams for the two principal directions when having a constant axial force (see Figure 03).

Figure 03 shows a quadrant of a three-dimensional My-Mz-N interaction diagram. Equation 5.39 is based on simplifying a horizontal section at NEd by using the 3D interaction diagram and generating a simplified My-Mz moment interaction diagram with the exponent a. In Figure 03, the actual My-Mz moment interaction diagram for the axial force NEd (horizontal section) is shown shaded in red. The simplified interaction diagram according to Equation 5.39 is also red-shaded for comparison. Figure 04 shows, depending on the exponent a, the distribution of the moment interaction applied in Equation 5.39 [1].

The advantage of this simplified approach according to Equation 5.39 [1] is that compression elements with biaxial eccentricities can also be designed quickly and easily by means of the known M-N interaction diagrams for uniaxial bending with axial force.

#### Exact Design of the Cross-Section with Biaxial Load Eccentricity

A precise design of a cross-section with axial force and biaxial bending requires an iterative calculation of the cross-section strains. The calculation of these cross-section strains is only possible with a calculation tool. The ultimate limit state design is fulfilled if the loading is within the Mres-N interaction diagram (grey-shaded area in Figure 03) or in the exactly determined Mz-M y moment diagram (red-shaded area in Figure 03). By precisely determining the limit curves, you can generate additional load capacities for the design.

#### Summary

For a biaxial load eccentricity, the standard allows different design variants depending on the load position. By respecting the boundary conditions of Equations 5.38a and 5.38b [1], the biaxial moment interaction can be neglected and the design can be carried out in the principal axis directions. If the limits mentioned above are exceeded, the moment interaction must be considered in the design. This is possible in a simplified way with the interaction formula according to Equation 5.39 [1] or by means of an accurate biaxial cross-section analysis. All described design approaches are possible with the RF-/CONCRETE Columns add-on module.

#### Reference

 [1] Eurocode 2: Design of concrete structures - Part 1‑1: General rules and rules for buildings; EN 1992‑1‑1:2004 +&nbso;AC:2010