Steel columns in simple construction are the most popular type of columns encountered in multi-storey steel buildings in practice. These are columns connected by nominally pinned joints and whose resistance to horizontal forces are provided by bracings in both directions. When steel frames are constructed with simple joints bending occurs only due to nominal eccentricities at the beam-to-column connections. It is on this basis that steel beams within braced multi-storey buildings are designed as simply supported members, simplifying the design process. Thus, the columns are designed only for a combination of axial forces and nominal moments from the eccentricity of the transferred forces.

The design of steel columns is well covered in most code of practices. Both BS-449 and BS-5950-1:2010 now superseded by BS-EN-1993-1-1 all contain simplified rules for columns in a simple construction. However, in BS EN 1993-1-1 the only option for members in combined bending and compression is the guidance given in clause 6.3.3(4). This is very complex compared to the procedure given in the former codes of practice. In view of this, the U.K NCCI document was developed to produce a simplified expression for the verification of columns in a simple construction.

**Verification of Column in Simple Construction**

The expression given in the NCCI is applicable for class 1, 2, or 3 cross-section provided the following criteria is satisfied.

- The section is hot rolled I, H or RHS section
- The bending moment diagram about each axis is linear
- Column is retrained laterally in both directions at each floor level but unrestrained between floors.

The expression is given as:

\frac { { N }_{ Ed } }{ { N }_{ min,b,Rd } } +\frac { { M }_{ y,Ed } }{ { M }_{ b,y,Rd } } +1.5\frac { { M }_{ z,Rd } }{ { M }_{ z,Rd } } \le 1.0

**Where**:

- N
_{Ed}is the applied axial load - M
_{y,Ed }is the applied bending moment in the y-y axis - M
_{z,Ed }is the applied bending moment in the z-z axis - N
_{b,Rd }is the axial buckling resistance - M
_{b,y,Rd }is the flexural buckling resistance about the y-y axis - M
_{z,Rd }is the bending moment resistance of the column in the z-z axis

**Applied Actions **

The applied axial load N_{Ed} can be determined using the tributary area method, or by summing the vertical reactions from the steel beams framing into the column. Axial load reduction factors may also be applied in estimating the applied axial actions.

Since columns in simple construction are nominally pinned, there are no significant moments applied. However, the NCCI document requires that a nominal moment be applied to columns in-order to account for eccentricity in steel connections. The value of these notional bending moments are given as:

{ M }_{ Ed,y }=\left[ \left( \frac { h }{ 2 } \right) +100 \right] \left( { R}_{ A }-{ R}_{ B } \right)

{ M }_{ Ed,z }=\left[ \left( \frac { b }{ 2 } \right) +100 \right] \left( { R}_{ A }-{ R}_{ B } \right)

**Where:**

- R
_{A}and R_{B}are the end reactions from the beams that are being supported by the column under consideration in the y or z-direction. - h is the depth of the column section
- b is the width of the column flanges

Having determined the bending moments, the moment is then distributed to the column above and that below the floor level. This is based on the proportion of the column section’s EI/L bending stiffness. If the ratio of this stiffness is less than 1.5 the bending moments are distributed equally to both upper and lower sections of the column.

**Axial Buckling Resistance Nb,Rd**

The axial buckling capacity of a steel column or a compression member is given as equation 6.47 of BS-EN-1993-1-1.

{ N }_{ min,Rd }={ \chi }_{ min }\frac { A{ f }_{ y } }{ { \gamma }_{ M1 } }

**Where:**

- f
_{y}is the yield strength of the steel section - A is the cross-sectional area of the steel section
- χ
_{min}is the minimum reduction factor due to buckling. This is determined for both axes and the minimum selected. - γ
_{M1}is the partial factor for the resistance of members subject to instability, which in the UK is set at 1.0

\chi=\frac { 1 }{ \left( \phi +\sqrt { { \phi }^{ 2 }-{ \lambda }^{ 2 } } \right) }

**Where:**

\phi =0.5\left[ 1+\alpha \left( \lambda -0.2 \right) +{ \lambda }^{ 2 } \right]

\lambda =\frac { { L }_{ cr } }{ i } \cdot \frac { 1 }{ { 93.9\varepsilon } }

- λ is the non-dimensional slenderness of the column
- L
_{cr}is the unrestrained length of the column along the axis that is being considered - i is the radius of gyration about the axis that is being considered
- ε is the coefficient for section classification
- α is the imperfection factor depending on appropriate buckling curve. This can be obtained from table 6.2 of BS EN 1993-1-

\varepsilon =\sqrt { \frac { 235 }{ { f }_{ y } } }

**Flexural Buckling Resistance about the Major Axis**

The flexural buckling resistance of a steel column is the same as that of a steel beam. This is described in Clause 6.3.2.1 (3) of BS EN 1993-1-1 in equation 6.55 as:

{ M }_{ b,y,Rd }={ \chi }_{ LT }\frac { { W }_{ y }{ f }_{ y } }{ { \gamma }_{ M1 } }

**Where:**

W _{y}is the major axis section modulus of the column section, which could be either elastic or plastic, depending upon its classification- χ
_{LT}is the reduction factor that takes into account lateral torsional buckling

{ \chi }_{ LT }=\frac { 1 }{ { \phi }_{ LT }+\sqrt { { \phi }_{ LT }-\beta { \lambda }_{ LT }^{ 2 } } }

{ \phi }_{ LT }=0.5\left[ 1+{ \alpha }_{ LT }\left( { \lambda }_{ LT }-{ \lambda }_{ Lt,0 } \right) +\beta { \lambda }_{ LT }^{ 2 } \right]

- The values of β and λ
_{LT,0}are defined as 0.75 and 0.4 respectively, as described in Clause NA.2.17 of the UK National Annex to BS-EN-1993-1-1 - α
_{LT}is the imperfection factor and is found in Table 6.3 of BS EN 1993-1-1, which relates to the steel column’s buckling curve. - λ
_{LT}is the non-dimensional slenderness of the column

{ \lambda }_{ LT }=0.9\lambda

**Flexural Resistance about the Minor Axis**

The flexural resistance of a steel column about the minor axis is the same as that of a steel beam. This is defined as:

{ M }_{ z,Rd }=\frac { { W }_{ z }{ f }_{ y } }{ { \gamma }_{ M1 } }

**Where:**

- W
_{z}is the major axis section modulus of the column section, which could be either elastic or plastic, depending upon its classification

Once the resistance to both axial forces and bending moments have been determined, the values are placed into the verification equation for columns in a simple construction.

Also see: Designing a Laterally Unrestrained Steel Beam

**Worked Example**

A steel column within a braced multi-storey steel building carries an ultimate axial compressive load of 3450kN at the ground floor. Four steel beams are connected to the column orthogonally, each of which exerts an out of balance moment. The ultimate shear loads are 600 kN and 375 kN in the y-y axis and 400kN and 300kN in the z-z axis. Assuming the column is the same size from the ground floor up till the topmost storey, with a floor-floor heights of 4m. Determine the adequacy of a 254x254x132 UC made from Grade S355 steel.

By inspection, this column is class one, therefore the rules for class one is valid.

**Design Actions**

The design axial compressive force load N_{Ed}=3450kN

{ M }_{ y }=\left[ \left( \frac { h }{ 2 } \right) +100 \right] \left( { R }_{ A }-{ R }_{ B } \right)

=\left( \frac { 276.3 }{ 2 } +100 \right) \cdot { 10 }^{ -3 }\times \left( 600-375 \right)

=53.6kN.m

Since the column is the same throughout the buiding, the ratio of the column stiffnesses is unity. Therefore the column moments will be shared equally.

{ M }_{ Ed,y }=\frac { 53.6 }{ 2 } =26.8kN.m

{ M }_{ z }=\left[ \left( \frac { b }{ 2 } \right) +100 \right] \left( { R }_{ A }-{ R }_{ B } \right)

=\left( \frac { 261.3 }{ 2 } +100 \right) \cdot { 10 }^{ -3 }\times \left( 400-300 \right)

=23.07kN.m

{ M }_{ Edz,z}=\frac { 23.07 }{ 2 }=11.54kN.m

**Axial Buckling Resistance**

By Inspection, buckling is more critical about the minor z-z axis. Therefore the minimum axial buckling resistance will be equal to the buckling resistance about the z-z axis.

{ N }_{ min,b,Rd }\quad ={ N }_{ b,z,Rd }\quad ={ \chi }_{ z }\frac { A{ f }_{ y } }{ { \gamma }_{ M1 } }

{ f }_{ y }={ R }_{ EH }=345N/{ mm }^{ 2 }

\varepsilon =\sqrt { \frac { 235 }{ { f }_{ y } } } =\sqrt { \frac { 235 }{ 345 } } =0.83

{ \lambda }_{ z }=\left( \frac { { l }_{ cr } }{ { i }_{ z } } \right) \cdot \left( \frac { 1 }{ 93.9\varepsilon } \right)

=\frac { 4000 }{ 66.9 } \cdot \frac { 1 }{ 93.9\left( 0.83 \right) } =0.77

The buckling curve is determined from table 6.27 of BS EN 1993-1-1. h/b =276.3/261.3 =1.0. Therefore buckling curve c applies. imperfection factor α=0.49

\phi =0.5\left[ 1+\alpha \left( \lambda -0.2 \right) +{ \lambda }^{ 2 } \right]

=0.5\left[ 1+0.49\left( 0.77-0.2 \right) +{ 0.77 }^{ 2 } \right]\\ =0.94

\chi =\frac { 1 }{ \phi +\sqrt { { \phi }^{ 2 }-{ \lambda }^{ 2 } } }

=\frac { 1 }{ \left( 0.94+\sqrt { { 0.94 }^{ 2 }-{ 0.77 }^{ 2 } } \right) } =0.68

{ N }_{ min,b,Rd }={ \chi }_{ z }\frac { A{ f }_{ y } }{ { \gamma }_{ M1 } }

=0.68\times \frac { \left( 168\times 100 \right) \times 345 }{ 1.0 } =3941kN

**Flexural Buckling Resistance about Major axis**

{ M }_{ b,y,Rd }={ \chi }_{ LT }\frac { { W }_{ y }{ f }_{ y } }{ { \gamma }_{ M1 } }

{ \lambda }_{ LT }=\quad 0.9\lambda \quad \\=\left( 0.9\times 0.77 \right) \quad =0.69

{ \phi }_{ LT }=0.5\left[ 1+{ \alpha }_{ LT }\left( { \lambda }_{ LT }-{ \lambda }_{ Lt,0 } \right) +\beta { \lambda }_{ LT }^{ 2 } \right]

=0.5\left[ 1+0.34\left( 0.69-0.4 \right) +0.75\times { 0.69 }^{ 2 } \right] \\=0.73

{ \chi }_{ LT }=\frac { 1 }{ { \phi }_{ LT }+\sqrt { { \phi }_{ LT }^{ 2 }-\beta { \lambda }_{ LT }^{ 2 } } }

=\frac { 1 }{ 0.73+\sqrt { { 0.73 }^{ 2 }-\left( 0.75\times { 0.73 }^{ 2 } \right) } } =0.91

{ M }_{ b,y,Rd }={ \chi }_{ LT }\frac { { W }_{ y }{ f }_{ y } }{ { \gamma }_{ M1 } }

=0.91\times \frac { \left( 1870\times { 10 }^{ 3 } \right) \cdot 345 }{ 1.0 } =645.15kN.m

**Flexural Resistance about minor axis**

{ M }_{ z,Rd }=\frac { { W }_{ y }{ f }_{ y } }{ { \gamma }_{ M1 } }

=\frac { \left( 878\times { 10 }^{ 3 } \right) \times 345 }{ 1.0 }=302.9kN.m

**Member Verification**

\frac { { N }_{ Ed } }{ { N }_{ min,b,Rd } } +\frac { { M }_{ y,Ed } }{ { M }_{ b,y,Rd } } +1.5\frac { { M }_{ z,Rd } }{ { M }_{ z,Rd } } \le 1.0

\frac { 3450 }{ 3941 } +\frac { 26.8 }{ 645.15 } +1.5\times \frac { 11.54 }{ 302.9 }

=0.97<1\quad o.k

Since the expression is satisfied, thus a **254 x 254 x 113 U.C** section is adequate and may be adopted.

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