1. Design of compression members

2. 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

3. 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

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

5. 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

6. Cross sectional resistance
Cross section resistance in compression is covered in Clause in EN
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

7. 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 = mm
b = 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)

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

9. Limit for Class 1 web = 33ε = 26.85
26.85 > 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

10. Buckling resistance members
Elastic compression member

11. 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

12. 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

14. 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

15. 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

16. 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 maybe used

17. 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 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

18. The basic formulations for the buckling curves are as given as below:
The buckling curve defined by EN 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.

19. 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

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

21. 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)

22. 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)

23. 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 )
Generally limited to cold-formed members
Flexural-torsional buckling mode(Clause )
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.

24. 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

25. 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

26. Figure 3 : Nominal buckling lengths Lcr for compression members

27. 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

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

29. 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

30. 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)

31. 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 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)

32. Figure 4 : a) Partial b) significant torsional and warping restraint from practical connections (Figure 6.13 of EN )

33. 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 EC3-1-1):
For Class 1, 2 , 3 cross section

34. 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 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

35. 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 EN )
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

36. 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

37. 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 The problem is as shown in sketch below:

38. 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 ( Table 3.1 EC3)

39. 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

40. 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
Clause
Clause

41. 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 , to carry a design axial compressive load of 1250kN.

42. 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 ( Table 3.1 EC3)

43. 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

44. Member buckling resistance :
effective length
Lcr,y = 1.0L = 1.0 X = 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
Table 6.1

45. Buckling reduction χ ;
Use 254 x 254 x 89 UC in grade S275 steel
Clause
Clause

46. 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 N/mm2 and a shear modulus of N/mm2. No allowance will be made for coatings in this example.
A = 5.55 cm2 iy = 3.92 cm IT = cm4
Aeff = 5.49 cm2 iz = 1.57 cm IW = 210 cm6
Iy = cm4 Wel,y = cm3 y0 = 3.01 cm
Iz = cm4 Wel,y = 3.83 cm3

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

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

49. 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
Table6.3 of EC3-1-3
Table 6.1 of EC3-1-1