13464x

001705

M.Eng. Milan Gérard

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 (M_G0, 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.

e_{0} = \frac{M_{{EdG}_{0}}}{N_{Ed}}

e₀	Eccentricity in relation to the center of gravity of a pure concrete section
M_EdG0	Design value of the bending moment in relation to the center of gravity of a pure concrete section
N_Ed	Design value of the acting axial force

Eccentricity in Relation to Center of Gravity of Pure Concrete Section

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

λ = \frac{l_{0}}{i}

l_{0} = β \cdot l

λ	Slenderness coefficient
l₀	Determined effective length
i	Radius of gyration of the uncracked concrete section
β	Buckling length coefficient
l	Free length

Slenderness Coefficient

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.

1 Predefined Values of Buckling Length Coefficient Around Y-Axis

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.

λ < λ_{\lim}

λ_{\lim} = \frac{20 \cdot A \cdot B \cdot C}{\sqrt{n}} according to French NA

A = \frac{1}{1 + 0.2 \cdot φ_{ef}} = 0.7 if φ_{ef} is unknown

B = \sqrt{1 + 2 \cdot ω} = 1.1 if ω is unknown

C = 1.7 - r_{m} = 0.7 if r_{m} is unknown

n = \frac{N_{Ed}}{A_{C} \cdot f_{cd}}

r_{m} = \{\begin{cases} 1, if unbraced elements in general \\ 1, if braced elements with moments of first order principally due to imperfections or transversal loads \\ \frac{M_{01}}{M_{02}} \leq 1, if other cases \end{cases}

λ	Slenderness criterion
λ_lim	Limiting slenderness
φ_ef	Effective creep coefficient
ω	Mechanical reinforcement ratio
r_m	Moment ratio
M₀₁, M₀₂	Algebraic values of the geometrically linear moments at both ends of the element

Slenderness Criterion

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.

φ_{ef} = φ (\infty, t_{0}) \cdot \frac{M_{0 E_{qp}}}{M_{0 Ed}}

φ_ef	Effective creep coefficient
φ(∞,t₀)	Final value of creep coefficient
M_0Eqp	Service moment of the first order under quasi-permanent combination of actions
M_0Ed	Ultimate moment of the first order under combination of design loads (including geometric imperfections)

Effective Creep Coefficient

2 Creep Parameters in Input Data

Walls and Isolated Columns of Braced Structures

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

e_{i} = θ_{i} \cdot \frac{l_{0}}{2}

θ_{i} = θ_{0} \cdot α_{h} \cdot α_{m}

θ_{0} = \frac{1}{200}

\frac{2}{3} \leq α_{h} = \frac{2}{\sqrt{l}} \leq 1

α_{m} = \sqrt{0.5 \cdot (1 + \frac{1}{m})}

e_i	Eccentricity due to imperfections
θ_i	Overall inclination of structure
θ₀	Basis value recommended by NA
α_h	Reduction coefficient relating to length
α_m	Reduction coefficient relating to the number of elements, where m is number of vertical elements contributing to the total effect

Eccentricity Due to Imperfections

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:

M_{{EdG}_{0}} = M_{Ed} + N_{Ed} \cdot ∆ e_{0}

∆ e_{0} = Max \{\begin{cases} 20 mm \\ \frac{h}{30} : required minimum eccentricity \end{cases}

M_EdG0	Bending moment
M_Ed	Design value of the bending moment
Δe₀	Required minimum eccentricity
h	Height of the straight section in the bending plane

Bending Moment

Calculation of Reinforcements 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 N_Ed and M_EdG0.

ν_{Ed} = \frac{N_{Ed}}{A_{c} \cdot f_{cd}}

A_{c} = b \cdot h

ν_Ed	Reduced axial force
A_c	Total area of the pure concrete section
b	Width of the straight section in the bending plane
f_cd	Design value of the concrete compressive strength

Reduced Axial Force

μ_{Ed} = \frac{M_{{EdG}_{0}}}{A_{c} \cdot h \cdot f_{cd}}

μ_Ed	Bending moment reduced in G₀

Bending Moment Reduced in G0

ρ = \frac{A_{s} \cdot f_{yd}}{A_{c} \cdot f_{cd}}

ρ	Mechanical percentage of reinforcement
A_s	Area of reinforcement
f_yd	Design yield strength of reinforced concrete steel

Mechanical Percentage of Reinforcement

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 the pure concrete of a braced structure element:
- Permanent:
  - N_g = 85 kN
  - M_g = 90 kN.m
- Varying:
  - N_q = 75 kN
  - M_q = 80 kNm
Materials:
- C 25/30 concrete
- Steel: S 500
Moment ratio at column base:
- |M01:| / |M02| = 1/3

3 RFEM Model with Modeled Permanent Loads

Material Characteristics

f_{cd} = α_{cc} \cdot \frac{f_{ck}}{γ_{c}}

f_cd	Design value of the concrete compressive strength
α_cc	Factor considering long term actions on the compressive strength
f_ck	Characteristic concrete compressive strength
γ_c	Partial safety factor for concrete

Design Value of Compressive Strength of Concrete

f_cd = 1 ⋅ 25 / 1.5 = 16.67 MPa

f_{y d} = \frac{f_{y k}}{γ_{s}}

f_yk	Characteristic yield strength of reinforcing steel
γ_s	Partial safety factor for reinforcing steel

Design Yield Strength of Reinforcing Steel

f_yd = 500 / 1.15 = 434.78 MPa

Ultimate Limit State

Loading of calculations in ultimate limit state:
M_Ed= 1.35 ⋅ M_g+ 1.5 ⋅ M_q
M_Ed= 1.35 ⋅ 90 + 1.5 ⋅ 80 = 241.50 kNm
N_Ed= 1.35 ⋅ N_g + 1.5 ⋅ N_q
N_Ed= 1.35 ⋅ 85 + 1.5 ⋅ 75 = 227.25 kN

4 Display of Governing Loads

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

5 User-Defined Buckling Coefficients in Input Data

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

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

Slendernesses
λ_y = 4.24 / 0.159 = 26.67 m
λ_z = 4.24 / 0.069 = 61.45 m

6 Display of Slenderness Values

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
λ_lim = (20 ⋅ 0.7 ⋅ 1.1 ⋅ 1.37) / √(0.103) = 65.74
λ_y< λ_lim ⟹ combined bending calculation in XZ plane
λ_z< λ_lim ⟹ calculation for simple compression in XY plane

7 Display of Slenderness Ratio Limits

As slenderness coefficients are lower than the limit values, buckling design of the part is useless; a combined bending calculation without taking into account the second-order effects is enough, under the eccentricity stresses below:
e₀= e₁ + e_i

Eccentricity due to computational loading
e₁= M_Ed / N_Ed
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
α_h = 2 / √6 = 0.816
α_m = √0.5 ⋅ (1 + 1 / 1) = 1
θ_i = 0.816 ⋅ 1 / 200 = 0.0041
e_i = 0.0041 ⋅ 4.24 / 2 = 0.0087 m

∆ e_{0} = Max \{\begin{cases} 20 mm \\ \frac{50}{30} \end{cases} = Max \{\begin{array}{l} 20 mm \\ 1.83 mm \end{array} = 20 mm

Straight Section with Symmetrical Reinforcement

8 Details of Eccentricity Calculation

Loading Applied to Center of Gravity of Pure Concrete Cross-Section

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

The minimum eccentricity is respected.
M_EdG0 = 227.25 ⋅ 1.072 = 243.61 kNm

9 Displaying Governing Moment Due to Eccentricity

Interaction Diagram for Rectangular Section with Symmetrical Reinforcement in Combined Bending

ν_Ed = (227.25 ⋅ 10^-3) / (0.24 ⋅ 0.55 ⋅ 16.67) = 0.103
μ_Ed = (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 ν_Ed, μ_Ed is accessible in the charts of interaction diagrams (Jean Perchat, Traité de béton armé, 3^rd 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.
A_s = (0.375 ⋅ 0.24 ⋅ 0.55 ⋅ 16.67) / (434.78) ⋅ 10⁴ = 18.98 cm²