Design of compression members

Sensitive to effects of Introduction Columns are vertical members used to carry axial compression load and due to their slenderness nature, they are prone to buckling The behaviour of column will depend on its slenderness as Prone to Sensitive to effects of High slenderness Medium slenderness Buckling Imperfection

Sensitive to effects of Stocky Columns are not affected by buckling and the strength is related to the material yield stress fy Nmax=Npl = Aefffy Stocky column Slender columns Intermediate columns Depend on Sensitive to effects of Depends on both Euler stress, σcr Material strength,fy

Common type of member Figure 1 : Typical column cross-sections

Typical section choices for a range of applications: Light trusses and bracing Angles (including compound angles back to back) Larger trusses Circular hollow section, rectangular hollow section, compound sections and universal columns Frames Universal Column (UC) fabricated sections e.g. reinforced UCs Bridges Box columns Power stations Stiffened box column

Cross sectional resistance Cross section resistance in compression is covered in Clause 6.2.4 in EN 1993-1-1. Only be applied as the sole check to members of low slenderness ( ≤ 0.2) For all other cases, check also need to be made for member buckling as defined in Clause 6.3 (EC3). The design compression force is donated by design normal force NEd . the design resistance of a cross section under uniform compression Nc,Rd is determined in similar manner to BS 5950:Part 1. The EC3 design expression for cross-section resistance under uniform compression -Ignore member buckling effect For Class 1,2 and 3 cross-sections For Class 4 cross-sections

Example 1 Question A 254 x 254 x 73 UC is to be used as a short ( ≤ 0.2) compression member. Calculate the resistance of the cross-section in compression, assuming grade S355 steel. Answer Section properties ( P363) h = 254.1 mm b = 254.6 mm tw = 8.6 mm tt = 14.2 mm r = 12.7 mm A = 9310 mm2 Yield strength, fy = 355 N/mm2 (Table 3.1 of EC3)

Cross-section classification (Clause 5.5.2) Outstand flanges (Table 5.2, Sheet 2) Limit for class 2 flange = 10ε = 8.14 8.14>7.77 flanges are class 2 Web internal compression part (Table 5.2,Sheet 1)

Limit for Class 1 web = 33ε = 26.85 26.85 > 23.29 web is Class 1 Overall cross-section classification is therefore Class 2. Cross-section compression resistance (Clause 6.2.4). for Class 1,2 or 3 cross-sections EC3 recommend

Buckling resistance members Elastic compression member

The load Ncr at which a straight compression member buckles laterally can be determined by finding a deflected position which is one of equilibrium. This position is given by In which  is the undeterminated magnitude of the central deflection, and that the elastic buckling load is The elastic buckling load Ncr and the elastic buckling stress

Also can be expressed in terms of the geometrical slenderness ratio L/i by The buckling load varies inversely as the square of the slenderness ratio L/i as shown in Figure below, in which the dimensionless buckling load Ncr/Ny is plotted against the generalized slenderness ratio In which

If the material ceases to be linear elastic at the yield stress fy, then the above analysis is only valid for This limit is equivalent to a slenderness ratio L/i of approximately 85 for a material with a yield stress fy of 275 N/mm2

Inelastic compression members the buckling of an elastic member of non-linear material, such as that whose stress-strain relationship is shown in below can be analysed by a simple modification of the linear elastic treatment Thus the flexural rigidity is reduced from EI to EtI,, therefore

Design members in compression Eurocode 3 (EC3) approach to determining the buckling resistance of compression member is based on same principles as that of BS 5950. Buckling resistance of members is covers in Clause 6.3(EC3) Guidance is provided for : Uniform compression members Uniform bending members Uniform members subjected to a combination of bending and axial compression No design expression for non-uniform member for calculating buckling resistance. However, noted that second-order analysis using the imperfections according to Clause 5.3.4 maybe used

Design compression force is denoted by NEd (design normal force) and must be shown to be less than or equal to the design buckling resistance of the compression member, Nb,Rd ( axial buckling resistance) Compression members with class 4 cross sections follows the provision of Clause 6.3.1 and the design buckling resistance should be taken as For Class 1,2 and 3 cross-sections For (symmetric) Class 4 cross-sections Where χ is the reduction factor for the relevant buckling mode

The basic formulations for the buckling curves are as given as below: The buckling curve defined by EN1993-1-1 are equivalent to those set out in BS5950:Part 1 (Table 24-with exception of buckling curve a0) The basic formulations for the buckling curves are as given as below: Where α is a constant ( imperfection factor) which shifts the resistance curve is non dimensional slenderness But χ ≤ 1.0 The advantage of a particular group of sections can be determined by assigning an appropriate value of α to them.

In EC3 (Clause 6.3.1), member slenderness or non-dimensional slenderness is defined as: Ncr is the elastic critical buckling force for the relevant buckling mode based on gross properties of the cross section The buckling mode that governs design will be that with the lowest critical buckling force Ncr. For class 1, 2 and 3 cross section For symmetric class 4 cross section

EN1993-1-1 defines five buckling curves, labelled ao,a,b,c and d as shown in Figure 2 (Figure 6.4 of EC3) Figure 2 : EN 1993-1-1 Buckling Curves

The shapes of these buckling curves are altered through the imperfection factor α and the five values of imperfection for each of these curve are given in Table 1 (Table 6.1 of EC3) To choice as to which buckling curve (imperfection factor) to adopt is dependent upon the geometry and material properties of the cross section an upon the axis buckling as shown in Table 2 (Table 6.2 of EC3). For compression members of stocky proportion ≤2.0 or NEd/Ncr ≤0.04, buckling effect may be ignored. Table 1 : Imperfection factors for buckling curves Buckling Curve a0 a b c d Imperfection factor α 0.13 0.21 0.34 0.49 0.76 Table 6.2 of EC3 is equivalent to allocation of strut curve (Table 23 of BS 5950: Part 1)

Selection of Buckling curve for a cross-section (Extract from Table 6 Selection of Buckling curve for a cross-section (Extract from Table 6.1 of EC3)

EC3 provides guidance for Flexural buckling mode(Clause 6.3.1.3) Standard hot-rolled and welded structural cross-section and it is predominant buckling mode and hence governs design in vast majority cases. Torsional buckling mode(Clause 6.3.1.4) Generally limited to cold-formed members Flexural-torsional buckling mode(Clause 6.3.1.4) Torsional for cold formed cross-section for two reason: cold-formed cross-section contain relatively thin material and torsional stiffness is associated with the material thickness cubed The cold forming process gives a predominance of open sections because these can be easily produced from flat sheet. open sections have inherently low torsional stiffness.

Flexural buckling Calculation of the non-dimensional slenderness for flexural buckling is given by Where Lcr is the buckling length of the compression member and is equivalent to the effective length LE in BS 5950 i is the radius of gyration about the relevant axis, determined using the gross properties of cross section (rx,ry in BS 5950) For class 1, 2 and 3 cross section For class 4 cross section

Table 2: Nominal buckling lengths Lcr for compression members (Table 24 BS 5950) End Restraint ( in the plane under consideration) Buckling length, Lcr Effectively held in position at both ends Effectively retrained in direction at both ends 0.7L Partially retrained in direction at both ends 0.85L Restrained in direction at one end Not restrained in direction at either end 1.0L One end Other end Buckling length, Lcr Effectively held in position at and restrained in direction Not held in position Effectively retrained in direction 0.7L Partially retrained in direction 0.85L Not restrained in direction 1.0L

Figure 3 : Nominal buckling lengths Lcr for compression members

Torsional and flexural torsional buckling To check possibility that the torsional or torsional flexural buckling resistance of a member maybe less than the flexural buckling resistance. Torsional buckling is pure twisting of a cross section and only occurs in centrally loaded struts which are point symmetric and have low torsional stiffness e.g cruciform section Torsional – flexural buckling is a more general response that occurs for centrally loaded struts with cross-sections that are singly symmetric and where the centroid and the shear center do not coincide E.g a channel section Whatever the mode of buckling of a member, the generic buckling curve formulation and the method for determining member resistance are common. Calculation of the non-dimensional slenderness for torsional and torsional flexural buckling resistance

Table 3: Buckling curve selection table (Table 6.3 of EN1993-1-3)

For class 1, 2 and 3 cross section Where Ncr=Ncr,TF but Ncr ≤ Ncr,T Ncr,TF is the elastic critical torsional-flexural buckling force Ncr,T is the elastic critical torsional buckling force The elastic critical buckling forces for torsional and torsional-flexural buckling for cross-sections that are symmetrical about(y-y) axis (i.e. where z=0) are given by For class 4 cross section

It is the torsional constant of the gross-section where i02=iy2+iz2+y02+z02 G is the shear modulus It is the torsional constant of the gross-section Iw is the warping constant if the gross-section iy is the radius of gyration of the gross-section about y-y axis iz is the radius of gyration of the gross-section about x-x axis lT is the buckling length of the member for torsional buckling y0 is the distance from the shear centre to the centroid of the gross cross section along the y axis z0 is the distance form the shear centre to the centroid of the gross cross section along z axis (Clause 6.33a of EC3-1-3)

Ncr,y is the critical force for flexural buckling about the y-y axis where Ncr,y is the critical force for flexural buckling about the y-y axis Guidance is provided in EN 1993-1-3 on buckling lengths for components with different degrees of torsional and warping restraint It is stated that for practical connections at each end lT/LT ( the effective buckling length divided by the system length) should be taken as 1.0 for connections that provide partial restraint against torsion and warping (Figure 4a) 0.7 for connections that provide significant restraint against torsion and warping (Figure 4b) (Clause 6.35 of EC3-1-3)

Figure 4 : a) Partial b) significant torsional and warping restraint from practical connections (Figure 6.13 of EN 1993-1-3)

Design of members in bending and axial compression Theoretically, all structural members be regarded as beam-columns, since the commons classifications of tension members, compression members and beams are merely limiting examples of beam-columns Members which subjected to combined bending and axial compression should satisfy (Clause 6.3.3 EC3-1-1): For Class 1, 2 , 3 cross section

For Class 4 cross section: Where: NEd, My,Ed and Mz,Ed are design values of the compression force and the maximum moments about the y-y and z-z axis along the member, respectively My,Ed, Mz,Ed are the moments due to the shift of the centroidal axis according to clause 6.2.9.3 and Table 4.5 χy and χz are the reduction factors due to flexural buckling χLT is the reduction factors due to lateral torsional buckling kyy,kyz,kzy,kzz are the interaction factors (Annex A and Annex B of EC3-1-1

Class 1 2 3 4 Ai A Aeff Wy W pl,y W eff,y Wz W pl,z W eff,z My,Ed Table 4.5 :values for NRk=fyAi, Mi,Rk=fyWi and Mi,Ed (Extract form Table 6.7 of EN1993-1-1) Class 1 2 3 4 Ai A Aeff Wy W pl,y W eff,y Wz W pl,z W eff,z My,Ed eN,y NEd Mz,Ed eN,z NEd

Buckling Resistance check summary Determine the axial load NED Choose a section and determine the class Calculate the effective length Lcr Calculate Ncr using the effective length Lcr and E and I which are section properties Calculate Determine α by first determining the required buckling curve from Table 6.1 (EC3) Calculate Ф by substituting in the values of α and Calculate χ by substituting in the values of Ф and Determine the design buckling resistance of the member

Example 2: Compression members (Rolled Universal Column design) Problem: Check the ability of a 203 x 203 x 52 UC in grade S275 steel to withstand a design axial compressive load of 1150kN over an unsupported height of 3.6m assuming that both ends of the member are pinned. Design to BS EN 1993-1-1. The problem is as shown in sketch below:

Partial factors: Geometric properties: A =66.3 cm2=6630mm2 i= 5.18cm = 51.8mm tf = 12.5mm cf/tf = 7.04 cw/tw = 20.4 Material properties; Yield strength fy = 275N/mm2 since tf ≤40mm UK NA to EC3 P363 (Steel building design: Design data) EN 10025-2 ( Table 3.1 EC3)

Check cross section classification under pure compression: Need only check that section is not class 4 (slender) For outstand flange cf/tf ε ε ≤ 14 For web cw/tw ε ≤ 42 Actual cf/tf ε ε = 7.04/0.92 = 7.62; within limit For web cw/tw ε = 20.4/0.92 = 20.2; within limit Section is not class 4 Cross- section compression resistance: Table 5.2 EC3 Clause 6.2.4

Member buckling resistance : Take effective length Lcr = 1.0L = 1.0 X 3600 = 3600 mm On the assumption that minor axis flexural buckling will govern, use buckling curve ‘c’ Use 203 x 203 x 52 UC in grade S275 steel Table 6.2 Clause 6.3.1.3 Clause 6.3.1.2 Clause 6.3.1.1

Example 3 : Pinned column with intermediate lateral restraints A 254 x 254 x 89 UC in grade S 275 steel is to be used as a 12.0m column with pin ends and intermediate lateral braces provided restraint against minor axis buckling at third points along the column length. Check the adequacy of the column, according to BS EN 1993-1-1, to carry a design axial compressive load of 1250kN.

Partial factors: Geometric properties: A = 113 cm2=11300mm2 iy= 11.2cm = 112mm iz= 6.55 cm = 165.5mm tf = 17.3mm cf/tf = 6.38 cw/tw = 19.4 Material properties; Yield strength fy = 275N/mm2 ; tf ≥ 40mm UK NA to EC3 P363 (Steel building design: Design data) EN 10025-2 ( Table 3.1 EC3)

Check cross section classification under pure compression: Need only check that section is not class 4 (slender) For outstand flange cf/tf ≤ 14ε For web cw/tw ≤ 42ε Actual cf/tf ε ε = 7.04/0.92 = 6.8; within limit For web cw/tw ε = 20.4/0.92 = 20.7; within limit Section is not class 4 Cross- section compression resistance: Table 5.2 EC3 Clause 6.2.4

Member buckling resistance : effective length Lcr,y = 1.0L = 1.0 X 12000 = 12000mm for buckling about the y-y axis Lcr,z= 1.0L = 1.0 X 4000 = 4000mm for buckling about z-z axis. Non-Dimension slendernesses: Buckling curves; For major axis buckling, use buckling curve ‘b’ For minor axis buckling, use buckling curve ‘c’ Clause 6.3.1.3 Table 6.1

Buckling reduction χ ; Use 254 x 254 x 89 UC in grade S275 steel Clause 6.3.1.2 Clause 6.3.1.1

Example 4: Member resistance in compression ( checking flexural, torsional and torsional-flexural buckling) Calculate the member resistance for a 100 x 50 x 3 plain channel section column subjected to compression. The column length is 1.5 m, with pinned end conditions, so the effective length is assumed to equal to the system length. The steel has a yield strength fy of 280N/mm2 a Young modulus of 210 000N/mm2 and a shear modulus of 81 000N/mm2. No allowance will be made for coatings in this example. A = 5.55 cm2 iy = 3.92 cm IT = 0.1621 cm4 Aeff = 5.49 cm2 iz = 1.57 cm IW = 210 cm6 Iy = 85.41 cm4 Wel,y = 17.09 cm3 y0 = 3.01 cm Iz = 13.76 cm4 Wel,y = 3.83 cm3

Calculate critical buckling load Flexural buckling-major (y-y) axis : Flexural buckling-minor (z-z) axis : Torsional buckling: Clause 6.2.3 of EC3-1-3

Torsional-flexural buckling: Torsional-flexural buckling is critical ( with Ncr = 114kN) Clause 6.2.3 of EC3-1-3

Non dimension slendeness (for torsional-flexural buckling mode) Selection of buckling curve and imperfection factor α For cold formed plain channel section, use buckling curve c For buckling curve c, α = 0.49 The member resistance of the 100 x 50 x 3 plain channel (govern by torsional-flexural buckling ) is 69.2 kN Clause 6.3.1.2 Table6.3 of EC3-1-3 Table 6.1 of EC3-1-1