# Effective length calculations (concrete column: EC2)

## Clear Height

The clear height is the clear dimension between the restraining beams at the bottom of the stack and the restraining beams at the top of the stack. The clear height may be different in each direction.

If, at an end of the stack, no effective beams or flat slab to include are found, then the clear height includes the stack beyond this restraint, and the same rules apply for finding the end of the clear height at the end of the next stack (and so on).

## Effective Length

The effective length, l_{0} is calculated automatically - you also have the ability to override the calculated value.

From EC2, cl. 5.8.3.2, the equations for calculating the effective length are as follows.

For stacks designated as "braced", the effective
length is given by^{1}:

l_{0} = 0.5 * l * √(1 + (k_{1} / (0.45 + k_{1}))) * √(1 + (k_{2} / (0.45 + k_{2})))

In addition Tekla Structural Designer imposes the following limits for stacks that are designated as braced:

5 ≤ l_{0} / l ≤ 1

For stacks designated as "bracing", the
effective length is the larger of^{2}:

l_{0} = l * √(1 + (10 * k_{1} * k_{2} / (k_{1} + k_{2})))

Or

l_{0} = l * (1 + (k_{1} / (1 + k_{1}))) * (1 + (k_{2} / (1 + k_{2})))

Where

k_{1} and k_{2} are the relative
flexibilities of rotational restraints at ends 1 and 2 respectively, in the
direction under consideration. Which way the ends are numbered is irrelevant to the
result. The program uses the bottom end of the stack as end 1 and the top end as end
2.

The value of *k*, which may refer to either
k_{1} or k_{2} depending on which end of the stack is being
examined, is defined by^{3}:

k = (θ / M) * (E * I / l)

Where

*M* is the moment applied to the restraining members by the buckling member or members,

θ is the rotation of the joint at the end of the stack considered for the bending moment
*M*,

*(E * I / l)* is the bending stiffness of the compression member or members considered to be buckling.

It is recommended to take "θ / M" for the beams as "l / (2 * E * I)" .

The standard approximation ^{4} for "θ / M" is between "l / (4 * E * I)" and "l / (3 * E * I)",
so to allow for cracking the value is increased. Also, "E * I / l" is the sum of the
stiffness of column stacks joining at the connection.

The above equation then becomes:

k = ∑(E * I / l)_{cols} / ∑(2 * E * I / l)_{beams}

If there are any adjacent stacks beyond the joint at the end of the restrained length under consideration, then it must be considered whether these adjacent stacks are likely to contribute to the deflection or restrain it. If the stiffness are similar then the stiffness of the adjacent stacks can be ignored, and the guidance in PD6687 suggests that this range of similarity of stiffness can be taken as 15% above or below the stiffness of the stack being designed. Therefore:

If

1.85 ≤ ∑((E *I/ l)_{stacks beyond this joint}) / (E * I/ l)_{stack under consideration} ≤ 1.15

Then

∑(E * I / l)_{cols} = (E * I / l)_{stack under consideration}

Else

∑(E * I / l)_{cols} = (E * I / l)_{stack under consideration} + ∑(E * I / l)_{stacks beyond this joint}

These stacks can be part of the same column length or another column length.

Note that as the restrained length may be multiple stacks, "E * I" for this stack are the values for the stack being designed, and l is the restrained length. For the stacks beyond the restraint, "E * I" are the values for the stack attached to the restraint, and l is the restrained length that the stack exists within.

Any beams framing into the end of the stack within 45 degrees of the axis being considered are said to be restraining beams for the stack in that direction.

There is a lower limit ^{5} for the value of k:

k ≥ 0.1

Additionally, Tekla Structural Designer imposes an upper limit:

k ≤ 20

For bracing stacks, a warning is displayed when the calculated value of k exceeds this limit.

## Fixed Column Base

k = 0.1 for fixed bases in Tekla Structural Designer. There is no clear guidance in EC2, but the Concrete Centre guidance suggests that this is suitable.

If you have set the bottom of the column to be "fixed" but the support as "pinned". The program will always assume that the support is fixed and therefore only ever consider the fixity applied to the column.

## Pinned Column End

In any situation where the end of a column anywhere in the structure is pinned, k = 20.

## No Effective Beams Found

If no effective beams are found to restrain the end of the stack in the direction in question, then the program will consider whether there is a flat slab restraining the stack at this end. If a flat slab is found it will either be considered as a restraint, or not, in each direction at each end of the stack - this is controlled by checking the option Use slab for stiffness calculation... located as a Stiffness setting in the column properties. If there are no effective beams and there is no flat slab (or any flat slab is not to be considered), then the program looks for the far end of the stack on the other side of the joint, and look at the restraints there, and so on until a restraint with an effective beam or flat slab to be considered is found.

If the stack is restrained by a flat slab, then the slab will be considered to act as a beam in this direction - note that it is one beam in the direction and NOT a beam on each side of the column.

If the stack is an end stack and there are no supports, beams or flat slabs considered to restrain the stack at this end in the direction, the end is therefore free in this direction and k = 20.

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*Footnotes*

^{1}EN 1992-1-1:2004 Section 5.8.3.2(3)

^{2}EN 1992-1-1:2004 Section 5.8.3.2(3)

^{3}EN 1992-1-1:2004 Section 5.8.3.2(3)

^{4}PD 6687-1:2010 Section 2.11.2

^{5}EN 1992-1-1:2004 Section 5.8.3.2(3)