10509x
001705
2021-02-05

Calculation of Concrete Columns Subjected to Combined Bending with RF-CONCRETE Columns

This article deals with elements concerning which the cross-section is subjected simultaneously to a bending moment, a shear force, and an axial compressive or tensile force. However, in our example we will not include loading due to shear force.

What is combined bending?

Combined bending is designated by the system (MG0, N) applied in a point C, called the center of pressure. The distance G0C is called the eccentricity of the external force in relation to the center of gravity G0 of the pure concrete section.

In combined bending, the value of the bending moment thus depends only on this point, where the reduction of the forces is carried out; here, it is G0.

The first thing to do in combined bending is to find the position of the center of pressure by calculating e0.

Considering Geometric Imperfections and Second-Order Effects in ULS

The analysis of elements and structures must take into account the unfavorable effects of any geometric imperfections in the structure, as well as deviations in the position of loads. Deviations in the sections' dimensions are normally taken into account by the partial safety factors for the materials.

Slenderness and Effective Length of Isolated Elements

Image 01 shows the possibility in RF-CONCRETE Columns of selecting the buckling length coefficient β by means of modeling the support conditions of isolated elements with constant section and the free length l.

Slenderness Criterion for Isolated Elements

It is assumed that second-order effects can be neglected if it is verified that the coefficient of slenderness is lower than the criterion of slenderness.

Considering Creep

The effect of creep must be taken into account in the second-order analysis, considering both the general conditions of creep and the application duration of different loads in a simplified manner by using an effective creep coefficient.


Walls and Isolated Columns of Braced Structures

In the case of isolated elements, the effect of imperfections can be taken into account as eccentricity ei.

Straight Cross-Sections with Symmetrical Reinforcement

To take into account deviations in the dimensions of cross-sections, the bending moment should be calculated in ULS:

Calculation of Steels Using Interaction Diagrams

The moment-normal force interaction diagrams are charts allowing for the rapid design or verification of straight cross-sections of which the shape and reinforcement distribution are determined in advance. The interaction diagrams are established only for the ultimate limit state. An interaction diagram is drawn using 2 curves constituting a continuous and closed contour called an interaction curve. The course of these curves is based on the equations of the resultant and the resulting moment, depending in particular on the following parameters:

  • Concrete and steel deformation diagrams
  • Concrete and steel stress diagrams

Thus, for the given cross-section (concrete, reinforcement, position of reinforcing steel), quantities are defined without dimension, based on the design internal forces NEd and MEdG0.

The last equation allows us to determine the necessary reinforcement section by interpolating the curve fields ρ of the interaction diagram, using the reduced orthonormal coordinate system (μ, υ).

Comparison of Theory with RF-CONCRETE Columns Add-on Module

Using a simple example, we compare the results in RF-CONCRETE Columns with the theoretical formulas described previously.

  • Loading applied to the center of gravity of pure concrete, of an element of a braced structure:
    • Permanent:
      • Ng = 85 kN
      • Mg = 90 kN.m
    • Varying:
      • Nq = 75 kN
      • Mq = 80 kNm
  • Materials:
    • C 25/30 concrete
    • Steel: S 500
  • Moment ratio at column base:
    • |M01:| / |M02| = 1/3

Material Characteristics

fcd = 1 ⋅ 25 / 1.5 = 16.67 MPa

fyd = 500 / 1.15 = 434.78 MPa

Ultimate Limit State

Loading of calculations in ultimate limit state:
MEd= 1.35 ⋅ Mg+ 1.5 ⋅ Mq
MEd= 1.35 ⋅ 90 + 1.5 ⋅ 80 = 241.50 kNm
NEd= 1.35 ⋅ Ng + 1.5 ⋅ Nq
NEd= 1.35 ⋅ 85 + 1.5 ⋅ 75 = 227.25 kN

Considering Geometric Imperfections Without Second-Order Effects in ULS

Geometric slenderness for isolated elements, considering the column inserted in a foundation block and restrained by a beam:
l0 = √2 / 2 ⋅ l = √2 / 2 ⋅ 6.00 = 4.24 m

Radius of gyration in plane parallel to side h = 55 cm
iy = h / √12 = 0.55 / √12 = 0.159 m

Radius of gyration in plane parallel to side h = 24 cm
iz = b / √12 = 0.24 / √12 = 0.069 m

Slendernesses
λy = 4.24 / 0.159 = 26.67 m
λz = 4.24 / 0.069 = 61.45 m

Limiting slenderness:
By default, the program takes into account the values according to the effects of creep for A, the initial reinforcements defined in RF-CONCRETE Columns for B, and the ratio of moments at the head and base of the analyzed member for C. However, it is possible to define these values yourself:
A = 0.7
B = 1.1
C = 1.7 - 1 / 3 = 1.37
n = (227.25 ⋅ 10-3) / (0.24 ⋅ 0.55 ⋅ 16.67) = 0.103
λlim = (20 ⋅ 0.7 ⋅ 1.1 ⋅ 1.37) / √(0.103) = 65.74
λy< λlim ⟹ combined bending calculation in XZ plane
λz< λlim ⟹ calculation for simple compression in XY plane

The slenderness coefficients being lower than the limit values, it is useless to check the part for buckling, and it is sufficient to have a calculation for combined bending without taking into account the effects of second order, under the following stresses of eccentricity:
e0 = e1 + ei

Eccentricity due to computational loading
e1 = MEd / NEd
e1 : Eccentricity due to computational loading
e1 = 241.50 / 227.25 = 1.063 m

Loading Corrected for Calculation of Combined Bending

Isolated column from braced structure:
θ0 = 1 / 200
αh = 2 / √6 = 0.816
αm = √0.5 ⋅ (1 + 1 / 1) = 1
θi = 0.816 ⋅ 1 / 200 = 0.0041
ei = 0.0041 ⋅ 4.24 / 2 = 0.0087 m


Loading applied to the center of gravity of the pure concrete cross-section:

e0 = e1+ ei ≥ Δe0
e0 = 1.063 + 0.0087 = 1.072 m

The minimum eccentricity is respected.
MEdG0 = 227.25 ⋅ 1.072 = 243.61 kNm

Interaction diagram for a rectangular section with symmetrical reinforcement in combined bending

νEd = (227.25 ⋅ 10-3) / (0.24 ⋅ 0.55 ⋅ 16.67) = 0.103
μEd = (243.61 ⋅ 10-3) / (0.24 ⋅ 0.552 ⋅ 16.67) = 0.201

The interaction diagram used to determine the required reinforcement according to the reduced forces νEd, μEd is accessible in the charts of interaction diagrams (Jean Perchat, Traité de béton armé, 3rd edition of LE MONITEUR, France, 2017).

In the graphical output, the found value is then interpolated between the interaction curves ρ = 0.35 and ρ = 0.40, resulting in ρ = 0.375.
As = (0.375 ⋅ 0.24 ⋅ 0.55 ⋅ 16.67) / (434.78) ⋅ 104 = 18.98 cm2

The difference of 0.10 cm² found for the reinforcement comes from the computer's precision on the interpolation of the values of the interaction diagram.


Author

Milan Gérard works at the Paris site. He is responsible for sales and provides technical support to our French-speaking customers.

Links
References
  1. Roux, J. (2007). Pratique de l'eurocode 2 - Guide d'application. Paris: Groupe Eyrolles.
  2. Perchat, J. (2017). Traité de béton armé - selon l'Eurocode 2, (3rd ed.). Antony Cedex: Groupe Moniteur.
  3. European Committee for Standardization (CEN). (2011). Eurocode 2: Design of Concrete Structures – Part 1-1: General Rules and Rules for Buildings, EN 1992-1-1:2011-01. Berlin: Beuth Verlag GmbH.


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