Columns are mostly used to transfer loads from superstructure to foundation or to transfer members. Although they are primarily compression members, they are often times subjected to moment due to eccentricity of loading or asymmetrical arrangement of supported beams or slabs. When a column is susceptible to moment, the critical design case is to arrange the loads in patterns that will generate maximum moment and axial load in the column. And according to clause 5.3.1 (7) of Eurocode 2 part 1, a column “is a member for which the section depth does not exceed 4 times its width and the height is at least 3 times the section depth”. If the member falls short of either of these criteria then it is considered as a wall.
Braced and Bracing Column
To effectively design a column, the ability to discern whether the column is braced or unbraced is paramount.
A column is said to be braced in a plane when it does not resist lateral loads or provide lateral stability in that plane. Lateral stability in such instance is meant to be provided by braces, shear walls, core walls, or other lateral load resisting elements incorporated in the structure.
A bracing column on the other hand is meant to give lateral stability to the structure in the plane under consideration.
Effective length of column
The degree of fixity or the type of constraints at each end of a column is also very important to the overall resistance of the column. Accommodating the effect of this end conditions brought about the concept of effective height of column.
EN. 1992.1.1.2004 gives, at least, three distinct methods to determine the effective length of column. The simplest of the approaches is to use fig 5.7 of the code which is reproduced here below. The diagram shows different isolated columns with varying boundary conditions and their associated effective length.
Another method and perhaps the most popular in various reference books is the formula method given in clause 5.8.3.2 (3) of the code. This is most suitable for columns in typical frames. These formulas consider the relative stiffness of the column to the beams framing into it. The formulas for braced and unbraced situations are as shown below:
For braced members:
$
_{\,\,l_0=0.5l\sqrt{\left( \text{1}+\,\,\frac{K_1}{\text{0.45}+\,\,K_1} \right) \left( \text{1}+\,\,\frac{K_2}{\text{0.45}+\,\,K_2} \right)}}
$
For Unbraced members, the larger of the two below formulas is used:
$
l_{o\,\,}=\,\,I\sqrt{\text{1}+\,\,10\left( \frac{K_{\text{1}}x\,\,K_2}{K_1+\,\,K_2} \right)}
$
or
$
l_{o\,\,}=\,\,I\left( \text{1}+\,\,\frac{K_1}{\text{1}+\,\,K_1} \right) \left( \text{1}+\,\,\frac{K_2}{\text{1}+\,\,K_2} \right)
$
Where: K1 and K2 are the rotational stiffness of the top and bottom ends of the column. K should be taken as zero for rigid restraint and ∞ for no restraint. However, since full rigid restraint is seldom in reality, the code suggests a limiting value of 0.1 is recommended for K1 and K2 when full rigidity is assumed.
There is another method given in clause 5.8.3.2 (7) where the effective length can be calculated using:
lo = Bw
β is a coefficient which depends on the member’s end conditions. The value of β can be obtained from table 12.1 of the code for different edge conditions. The table is specifically applicable to walls. However, for columns, β can be generally assumed to be 1 and specifically assumed to be 2 for cantilever columns.
The effective height of a column is very important as it is a factor that determines the slenderness of the column or the susceptibility of the column to buckling. In designing against buckling, the code categorizes columns into two:
- Non-Slender Column
- Slender Column
Slenderness and Limiting Slenderness
According to clause 5.8.3.2 (1), the slenderness of a column can be estimated using:
λ = lo/I
The slenderness ratio obtained using the above equation is checked against the limiting slenderness. The limiting slenderness is calculated using: λlim = (20 X A X B X C/n)
The definition of each variable in the equation is reproduced below as defined in equation (5.13N) of EC 2.
A = 11(1+0,2øef) (if øef is not known, A = 0.7 may be used)
B = (1 +2w)0.5 ( if (w) is not known, B = 1.1 may be used)
C = 1.7 – rm (If rm is not known, C = 0.7 may be used)
For unbraced column and column which much of the moment on it is due to geometric imperfections then C should be adopted as 0.7
øef = effective creep ratio; see 5.8.4 of EC2
w = Asfyd;/Acfcd; mechanical reinforcement ratio
As = total area of longitudinal reinforcement
n = NEd /Acfcd; relative normal force.
rm = Mo1/Mo2; Moment ratio. Mo1 and Mo2 are the first-order end moments. IMo2I ≥ IMo1I
When the slenderness ratio is less than the limiting slenderness, the column is considered short and second order effect is neglected. However, when the slenderness ratio is greater than the limiting slenderness, the column is treated as slender and the slenderness effect is to be considered.
Critical Design Moment in columns
In order to analyze column to obtain critical moment to be adopted for designing the member, the code suggests four methods which are:
- The general method.
- Method based on Nominal Stiffness.
- The moment magnification method
- Method based on Nominal Curvature.
The fourth method is further dealt with in this article as this is similar to that obtainable in previous British standards. In order to clearly categorize columns based on the computation of the design moment that will be adopted in sizing them assuming the moment is about a single axis, columns are better categorize into two, which are:
- Non-Slender Column
- Slender column
Non-Slender Column
The design of a short column is independent on whether it is braced or unbraced. Short columns are likely to fail by crushing so the axial load on them is always preponderance. As for the moment acting on them it is pretty straightforward.
Since MED = MoED + M2
M2 is the moment due to second-order effect and it is always insignificant in short columns. Hence, we are left with MoED which is the larger of the moments acting on the two ends of the column plus the effect of geometric imperfections. Hence, the design moment (MED ) to be adopted in designing a short column is MoED
MoED = Mo2 + e1 X Ned
Mo2 is the greater between the moment at the top and bottom of the column. ie: Mo2 = Max(IMtop I, IMbottom I.
Conversely, Mo1 is the lesser between the moments at the top and bottom of the column. Ie: Mo1 = Min(IMtop I, IMbottom I. However Mo2 is mostly used in sizing a column rather than Mo1 as it is more critical.
Hence MoED = Max(IMtop I, IMbottom I + e1 X Ned
e1 which is the eccentricity due to imperfection given by lo/400. The imperfection must not be lesser than the minimum eccentricity eo prescribed by the code. The minimum eccentricity is the greater between h/30 and 20mm. Whenever the moment due to minimum eccentricity is greater than the moment due imperfection then the equation becomes:
MoED = Mo2 + e0 X Ned
MoED = Max(IMtop I, IMbottom I + e0 X Ned
Click here to read a worked example on the design of non-sender column to Eurocode 2
Braced Slender Column
The critical moment of a slender column is obtained by adding the first-order moment, which includes the effect of imperfections, to the second-order moment induced by the slenderness of the column. This is given in clause 5.8.8.2(1) of the code as:
MED = MoED + M2
MoED and M2 are first-order moment with effect of geometric imperfection, and second-order moment respectively.
M2 = NED + e2
where: NED is the design axial force.
e2 is the deflection of the column
e2= 1/r lo2 / c
1/r is the curvature
lo is the effective length
C is the which depends on the curvature distribution.
The curvature (1/r) =Kr Ko 1/ro
Axial load correction factor (Kr)
Kr = (nu – n)/ (nu -nbal) ≤ 1
nu = 1 + ω
ω = Asfyd/Acfcd
Creep Factor (Kϴ)
Kϴ = 1+ B ϕef ≥ 1
B = 0.35 + fck/200 – ⅄/150
effective creep ratio (ϕef) = This can be easily gotten by making ϕef the subject of formular in Equ 5.13N of Eurocode 2, when A is assumed to be 0.7)
The critical moment which should be adopted as the design moment (MED) should be the greatest of:
a) Mo2 + e0 X Ned
b) Moe + M2
c) Mo1 + e0 X Ned + M2/2
Moe is the equivalent first-order moment for columns without load applied between their ends. In such a column Moe is meant to replace the differing end moments which are Mo1 and Mo2
Equation 5.32 of the code renders Moe as Moe = 0.4Mo2 + 0.6Mo1 ≥ 0.4Mo2 . Where Mo1 and Mo2 are the small and large moment in the column assuming the column is bent about a double curvature.
Click here to read a worked example on the design of sender column to Eurocode 2
Bracing Slender Column
Eurocode 2 barely mention any guidance about analyses or design of bracing slender column other than the provision for its effective length. BS 8110 can be referred to for succor in computing the design moment. To obtain the design moment, the additional moment due to the second-order effect should be added to moments at its ends. This makes the equation MED = MoED + M2 to be still very much relevant.
However, MoED here can only be the first-order moment at the stiffer end plus the effect of imperfection. Below is an image from BS8110:1:1997 that perfectly captures the moment on slender columns.
If a Column is Subjected to bending about both axis (Biaxial Bending)
For most columns biaxial bending is often not always critical as large moment is both directions are seldom the case. In clause 5.8.9(3), the code gives criteria which when they are met then biaxial bending design should be ignored.
Biaxial bending should be ignored in a column when:
λy/λz ≤ 2 and λz/λy ≤ 2
And
(ey/heq)/(ez/beq) ≤ 0.2 or (ez/beq)/(ey/heq) ≤ 0.2
There is no exact method of designing biaxial column in the Eurocode but only checks which the member must conform with after it must have been designed with whatever method catches the fancy of the designer. The check that must be satisfied is given by equation (5.39) of the code where the moments acting about both axes is related to moment of resistance about both axes as thus:
$
\left( \frac{M_{Edz}}{M_{Rdz}} \right) ^{\alpha}\,\,+\,\,\left( \frac{M_{Edy}}{M_{Rdy}} \right) ^{\alpha}\,\,\leqslant \,\,1
$
Many reference books in the UK maintain biaxial column should be designed according to BS 8110 -1-1: 1997. BS 8110 -1-1 1997 permits biaxial columns to be designed to withstand an increased moment about one axis. The axis with which the moment will be increased is determined as follows:
If Mz/h’≥ My/b’ then the increased moment will be about z axis. The increased moment should be calculated thus: Mz‘ = Mz + βh’/b’ My
If Mz/h’ ≤ My/b’ then the increased moment will be about y axis. The increased moment should be calculated thus: My‘ = My + βh’/b’ Mz
Design of Columns.
In most regular columns, serviceability criteria such as deflection and cracking are not critical so the columns are mostly designed for Ultimate Limit State of strength.
Most columns are practically always subjected to both axial stress and bending stress even if the bending stress is nominal. There are different methods that can be used in designing columns some of which are listed below:
- Approximate equations
- Column design Charts
- Using M-N Interaction Diagram or Capacity Curve.
Reinforcement Detailing
- A minimum of four bars is required in a rectangular column while a minimum of six bars is required in a circular column.
- The longitudinal bar diameter must not be less than 12mm
- The minimum area of steel is given by As = 0.1NED /0.87fyk ≥ 0.002Ac
- Maximum Area of steel is Asmax < 0.04Ac
- Maximum Area of steel is Asmax < 0.08Ac
- Minimum size of links equal 0.25 x the diameter of the compression reinforcement but not less than 6mm
- Every main bar in a corner should be firmly held by a link.
- Maximum spacing should not be greater than smallest longitudinal bars x 20 or the smaller of sides of the column’s cross-section or 400mm
References:
BS EN 1992-1-1:2004: Design of concrete structures Part 1: General rules and rules for building
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