# Column stack classification (concrete column: EC2)

## Slenderness ratio

The slenderness ratio, λ of the restrained length about each axis is calculated as follows:

λ = l_{0} / i = l_{0} / √(I / A)

Where

l_{0} is the effective height of the stack,

i is the radius of gyration of the stack section about the axis under consideration,

I is the second moment of area of the stack section about the axis,

A is the cross-sectional area of the stack section.

The slenderness ratio λ is then checked against the limiting slenderness ratio λ_{lim} in each direction. If the slenderness is less than this limit, then the member is short and slenderness effects are ignored, otherwise it is slender.

## Limiting slenderness ratio

λ_{lim} = 20 * A * B * C / √n

Where

A = 1 / (1 + (0.2 * φ_{ef})) ≥ 0.7

B = √(1 + (2 * ω)) ≥ 1.1

C = 1.7 - r_{m}

Where

φ_{ef} is the effective creep ratio,

ω = A_{s} * f_{yd} / (A_{c} * f_{cd}),

f_{yd} is the design yield strength of the reinforcement,

f_{cd} is the design compressive strength of the concrete,

A_{s} is the total area of longitudinal reinforcement,

n = N_{Ed} / (A_{c} * f_{cd}),

N_{Ed} is the design axial force between restrained floor levels in this direction,

r_{m} = M_{1.1} / M_{2.1}

M_{1.1} and M_{2.1} are the first order moments at the ends of the stack about the axis being considered, with |M_{2.1}| ≥ |M_{1.1}|.

If M_{1.1} and M_{2.1} cause tension in the same side of the stack then r_{m} is positive and C ≤ 1.7. If the converse is true then the stack is in double curvature, and it follows that r_{m} is negative and C > 1.7.

For braced stacks in which the first order moments arise only from transverse loads (lateral loading is significant) or imperfections (M_{imp.1} > |M_{2.1}|), C must be taken as 0.7,

For

bracing stacks, C must be taken as 0.7,

For restrained lengths encompassing more than one stack, C is taken as 0.7.

The effective creep ratio, φ_{ef}, is derived as follows:

f_{cm} = f_{ck} + 8 (N/mm^{2})

h_{0} = 2 * A_{g} / u

Where

u is the section perimeter in contact with the atmosphere (assumed to be the full section perimeter),

A_{g} is the gross section area.

α_{1} = (35 / f_{cm})^{0.7}

α_{2} = (35 / f_{cm})^{0.2}

α_{3} = (35 / f_{cm})^{0.5}

If f_{cm} ≤ 35 N/mm^{2},

β_{H} = (1.5 * (1 + (1.2 * RH))^{18} * h_{0}) + 250 ≤ 1500

Else,

β_{H} = (1.5 * (1 + (1.2 * RH))^{18} * h_{0}) + (250 * α_{3}) ≤ 1500 * α_{3}

Where

RH is the relative humidity, which is set under Design parameters in the column properties.

β_{c}( t , t_{0} ) = ((t - t_{0}) / (β_{H} + t - t_{0}))^{0.3}

β_{t0} = 1 / (0.1 + t_{0}
^{0.2})

β_{fcm} = 16.8 / √f_{cm}

Where

t_{0} is the age of column loading and defaults to 14 days, if required it can be changed under Design parameters in the column properties.

If f_{cm} ≤ 35 N/mm^{2},

φ_{RH} = 1 + ((1 - (RH / 100)) / (0.1 * h_{0}
^{1/3}))

Else,

φ_{RH} = (1 + (((1 - (RH / 100)) / (0.1 * h_{0}
^{1/3})) * α_{1})) * α_{2}

Then,

φ_{0} = φ_{RH} * β_{fcm} * β_{t0}

φ( ∞ , t_{0}) = φ_{0} * β_{c}( ∞ , t_{0} )

If φ( ∞ , t_{0}) ≤ 2 and λ < 75 and M_{max.1} / N_{Ed} ≥ h and ω ≥ 0.25,

φ_{ef} = 0

Else

φ_{ef} = φ( ∞, t_{0}) * R_{PL}

Where

M_{max.1} is the largest first order moment in the restrained length in this direction,

N_{Ed} is the design axial force in the restrained length in this direction,

R_{PL} is the permanent load ratio.

You are required to supply a value for the permanent load ratio which is located under Design parameters in the column properties. A default of 0.65 has been assumed, but you are advised to consider if this is appropriate and adjust as necessary.

Tekla Structural Designer assumes that t ∞ (t-infinity) is equal to 70 years (25550 days).

## Pre-selection of Bracing Contribution

The significant parameter within the slenderness criteria is a choice of how a wall, or column, is contributing to the stability of the structure.

In-plane direction, a wall is usually considered to be a bracing member. Out-of-plane direction, a wall is usually considered to be braced by other stabilizing members. These are the default settings but can be edited.