# 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 Without Considering Biaxial Moment Interaction

According to 5.8.9 (2), EN 1992-1-1 [1], it is allowed to design a column with biaxial load eccentricity separately in both principal axis directions without taking into account the moment interaction, if they comply with the limits of Equations 5.38a and 5.38b [1]. The equations are described in the linked article above. These limits and the design approach are based on the fact that one of the two load eccentricities represents the dominant value and the second one represents the subordinate value. When designing in the separate principal axis directions, the additional eccentricity from imperfection must be considered only in the dominant direction, which means in the governing direction. Figure 01 shows which areas do not have to be considered for the biaxial bending. If the eccentric axial force is in the shaded areas, you can design the column separately in both principal directions. Please note that that the load eccentricities have to be considered according to the second-order analysis in the respective principal axis directions. This is also shown in Figure 01. Points A and B in Figure 01 symbolize two examples of a possible load position where the influence of eccentricity according to the second-order analysis has different effects. Without considering the second-order analysis (e_{2}), both points are located in shaded and allowable areas where the biaxial bending design can be neglected. When considering the eccentricity according to the second-order analysis, the biaxial load eccentricity is reduced for point A, whereas for load position B, biaxial bending is increased and the load is shifted out of the allowable range.

Figure 01 - Biaxial Load Eccentricities in Cross-Section

#### 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

M_{Ed,z/y} = design moment about the corresponding axis, including nominal second order moments

M_{Rd,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 - Exponent a for Equation 5.39

Figure 02 shows the exponent a as a function of the ratio N_{Ed}/N_{Rd}. N_{Rd} is the design value of the centric axial resistance and can be determined with N_{Rd} = A_{c} ⋅ f_{cd} + A_{s} + f_{yd}. A_{c} represents the gross cross-section area, A_{s} the longitudinal reinforcement area, and f_{cd} and f _{yd} the design strengths of the used materials.

When using Equation 5.39 [1], please note that the two bending resistances M_{Rdy} and M_{Rdz} 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 - D Interaction Diagram

Figure 03 shows a quadrant of a three-dimensional M_{y}-M_{z}-N interaction diagram. Equation 5.39 is based on simplifying a horizontal section at N_{Ed} by using the 3D interaction diagram and generating a simplified M_{y}-M_{z} moment interaction diagram with the exponent a. In Figure 03, the actual M_{y}-M_{z} moment interaction diagram for the axial force N_{Ed} (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].

Figure 04 - Simplified Distribution of the Moment Interaction According to Equation 5.39

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 M_{res}-N interaction diagram (grey-shaded area in Figure 03) or in the exactly determined M_{z}-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.

#### Keywords

Biaxial Load eccentricity Eccentricity Column design Equation 5.39 Nominal curvature method Moment interaction Interaction diagram Reinforced concrete column Compression element

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