Design of Concrete Columns Subjected to Axial Compression with RFCONCRETE Columns
Technical Article
This article compares the design to the one in the referenced article: Design of Concrete Columns Subjected to Axial Compression with RFCONCRETE Members. It is, therefore, about taking exactly the same theoretical application carried out in RFCONCRETE Members and reproducing it in RFCONCRETE Columns. Thus, the objective is to compare the different input parameters and the results obtained by the two addon modules for the design of columnlike concrete members.
Theoretical Application
Axial compression applies if it is assumed that secondorder effects (imperfections, asymmetry, etc.) can be neglected while respecting in particular the slenderness criterion, which depends on various parameters (slenderness coefficient, limiting slenderness, effective length).
Then, under the single loading of a normal force N_{Ed}, the force that can be balanced by the concrete crosssection corresponds to its maximum loadbearing capacity for compression, which directly depends on its section and its design resistance. The reinforcement will balance the rest of the axial compressive load.
Application of the Theory with the RFCONCRETE Columns addon module
In this article, we will analyze the results obtained automatically for the reinforcement calculation.
The parameters remain the same and are listed below:
 Permanent loads: N_{g} = 1 390 kN
 Variable loads: N_{q} = 1,000 kN
 Column length: l = 2.1 m
 Rectangular crosssection: width b = 40 cm / height h = 45 cm
 The column's selfweight can be ignored.
 Column not integrated into bracing.
 Concrete strength class: C25/30
 Steel: S 500 A for inclined graph
 Diameter of longitudinal reinforcement: ϕ = 20 mm
 Diameter of transverse reinforcement: ϕt = 8 mm
 Concrete cover: 3 cm
Real CrossSection to Be Calculated
Since it is impossible in RFCONCRETE Columns to optimize the crosssection height, the section's real height h is directly modified and set to 45 cm.
Image 02 shows the steps to change the height of the rectangular crosssection in RFCONCRETE Columns.
Material Properties
The formulas for the materials' strength and strain are described in detail in the technical article mentioned above.
Total area of pure concrete section
A_{c} = b ⋅ h = 0.40 ⋅ 0.45 = 0.18 m²
Design value for compressive strength of concrete
f_{cd} = 16.7 MPa
Relative compression strain for maximum stress
ε_{c2} = 2‰
Design yield strength of reinforcing steel
f_{yd}= 435 MPa
Limit strain in reinforcement
ε_{ud} = 2.17‰
Stress in reinforcement
σ_{s} = 400 MPa
In order to verify the material settings in RFCONCRETE Columns, Image 03 shows the expected stresses and strains for the concrete and the required reinforcement.
Ultimate Limit State
Ultimate limit state design loads
N_{Ed} = 1.35 ⋅ N_{g} + 1.5 ⋅ N_{q}
N_{Ed} = 1.35 ⋅ 1390 + 1.5 ⋅ 1000 = 3.38 MN
N_{Ed} ... design value of acting axial force
SecondOrder Effects not Taken into Account in ULS
As the model is identical for this article and the one that serves as a basis for comparison, we have modeled the same column restrained at the base and free at the head to be able to apply the load correctly at the column head. However, we consider that the column is still fixed at the head to some beams, and for this we have applied an effective length factor to the column, which allows for modifying the column's slenderness value.
Effective length factor according to EN 199211  5.8.3.2 (3)  Formula 5.15
k_{cr} = 0.59
Slenderness according to EN 199211  5.8.3.2 (1)  Formula 5.14
λ_{z} = 10.73 m
Limiting slenderness according to EN 199211  5.8.3.1 (1)  Formula 5.13N
n = 1.125
λ_{lim} = 20 ⋅ 07. ⋅ 1.1 ⋅ 0.7 / √1.125 = 10.16 m
λ_{z} > λ_{lim} → The condition is not fulfilled.
However, we are still going to calculate the column with regard to simple compression because, the difference being small, we see below that with the mechanical reinforcement ratio the condition will be respected. For this, Image 05 describes how to deactivate the possibility of having buckling around each axis of the crosssection in RFCONCRETE Columns.
LoadBearing CrossSection
Equilibrium force of concrete
F_{c} = A_{c} ⋅ f_{cd} = 0.40 ⋅ 0.45 ⋅ 16.7 = 3 MN
Equilibrium force of reinforcement
F_{s} = N_{Ed}  F_{c} = 3.38  3 = 0.38 MN
We deduce the corresponding reinforcement area:
Reinforcement area
A_{s} = F_{s} / σ_{s} = 0.38 / 400 ⋅ 10^{4} = 9.5 cm²
By having set the reinforcing steels for a diameter of 20 mm in RFCONCRETE Columns, the provided reinforcement determined automatically by the addon module is 4 members, with a distribution in the corners as requested, i.e., 1 HA 20 per corner, which results in the following reinforcement area:
A_{s} = 4 ⋅ 3.142 = 12.57 cm²
Mechanical reinforcement ratio
ω = (A_{s} ⋅ f_{yd}) / (A_{c} ⋅ f_{cd}) = 0.182
Final check of limiting slenderness
λ_{lim} = (20 ⋅ 0.7 ⋅ √(1 + 2 ⋅ 0.182) ⋅ 0.7) / √1.125 = 10.79 m
λ_{z} < λ_{lim} → The slenderness criterion is fulfilled.
Author
M.Eng. Milan Gérard
Sales & Technical Support
Milan Gérard works at the Paris site. He is responsible for sales and provides technical support to our Frenchspeaking customers.
Keywords
Eurocodes Compression Reinforcement Slenderness
Reference
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