Topic 8 - steel columns

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Topic 8 Columns Design

Prof Dr Shahrin Mohammad

Structural Steel and Timber Design SAB3233

Structural Steel Design

Topic 1 -‐ Overview

Topic 4 – Design of steel structures (BS EN 1993)

Topic 3 –AcKons on

Structures (BS EN 1991)

Topic 5 – Cross-‐secKon

classificaKon

Topic 2 -‐ Basis of Structural Design (BS EN 1990)

Topic 6 – Laterally restrained beams

Topic 7 – Laterally unrestrained beams

Topic 8 – Columns Design

Topic 9 – Trusses

Topic 10 -‐ ConnecKons

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Columns subjected to axial load



Mx,t My,t

x

y

Mx,b

My,b Squashing, normally occurs in short column

Overall flexural buckling

Torsional buckling

Local buckling

NEd NEd NEd NEd NEd

NEd = Design value of compression force

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5

! compression members subject to §  axial compression only §  no bending

! however in pracKcally real columns are subject to §  eccentriciKes of axial loads §  transverse forces

! the treatment disKnguishes between §  stocky columns, and §  slender columns

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Stocky columns   The characterisKcs of stocky columns are

•  very low slenderness •  unaffected by overall buckling

  The compressive strength of stocky columns is

•  dictated by the cross-‐secKon •  a funcKon of the secKon classificaKon

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Slender Steel Columns

  Slender columns present a quasi elasKc buckling behaviour

  Euler criKcal stress λ = Lcr / r, where r is radius of gyraKon

Lcr is the buckling length

2

2

λπ

σE

cr =

Failure by yielding

Failure by buckling

Euler buckling curve

λ

σ

fy

λ1

Euler buckling curve and modes

of failure

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Behaviour of real steel columns

q  columns of medium slenderness are very sensiKve to the effects of imperfecKons

q  inelasKc buckling occurs before the Euler buckling load due to various imperfecKons §  iniKal out-‐of-‐straightness §  residual stresses §  eccentricity of axial applied loads §  strain-‐hardening

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Effects of imperfecKons

•  structural imperfecKons most important for intermediated columns

•  this represents most pracKcal columns

•  lower bound curve is obtained from a staKsKcal analysis of test results

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Effect of imperfecKons in relaKon to slenderness

  Slender column ⚈  largely unaffected by imperfecKons ⚈  ulKmate failure load ≈ Euler load (Ncr) ⚈  independent of the yield stress

  Intermmediate column ⚈  imperfecKons important ⚈  failure load less than Euler load ⚈  out-‐of-‐straightness and residual stresses are the most significant imperfecKons

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Residual stresses pa_erns

Typical residual stress pa_ern

y

y

y

0 , 3 f

0 , 2 f

0 , 2 f

∼

∼

∼ compression

tension

compression

+ =

σ R N / A σ n f y < σ n f y

f y

o r =

reaching

CombinaKon with axial stresses

•  combined with axial stresses cause yielding •  effecKve area reduced

Modify

IniKal out-‐of-‐straightness

! induces bending moments

Yielded zones

Combined effect of imperfec0ons and axial load

! bending stress σB ! residual stress, σR ! applied axial stress, NEd/A

+ +

P / A σ R

=

σ B σ max

NEd

NEd

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Buckling resistance in axial compression

The design buckling resistance of a compression member

Nb.Rd = χAfyγM1

1.

M

yeff

Rdb

fAN

γχ=

for Class 1, 2 and 3 cross secKon

for Class 4 cross secKon

where χ a reducKon factor and is related to the reference slenderness Buckling curves plo_ed as χ versus reference slenderness raKo

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European buckling curves (Cl 6.3.1.2)

Ncr is the elastic critical force for the relevant buckling mode based on the gross cross sectional properties.

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European buckling curves

Based on experiment more than 1000 tests secKon Range of slenderness raKos between 55 and 160

•  Based on a half sine-‐wave geometric imperfecKon = L/1000

•  residual stresses related to secKon type

•  4 curves apply to different cross-‐secKon types corresponding to different values of the imperfecKon factor α

Modify

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ImperfecKon factor a

•  α depends on §  the shape of the column cross-‐secKon §  the direcKon of buckling (y or z axis) §  the fabricaKon process (hot-‐rolled, welded or cold-‐formed

•  imperfecKon factors given in Table 6.1

Non dimensional slenderness, ___λ

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SelecKon of appropriate buckling curve

Table 6.2 helps with the selection of the appropriate buckling curve

Modify

Example 1 : Design of an axially loaded column

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Columns subjected to combined

bending and axial load



Mz,t

y

z

My,t Mz,t

My,b

Mz,b

y

z

My,t Mz,t

My,b

Mz,b

y

z Pin

connecKon Pin

connecKon Pin

connecKon

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Mz,t

y

z

My,t Mz,t

My,b

Mz,b

y

z

My,t Mz,t

My,b

y

z Mz,b

My,b

Mz,b

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simple construcKon

‘Simple construcKon’ is commonly used for the design of mulK-‐storey buildings

•  Beams are designed as simply supported

•  Columns are designed for nominal moments arising from the eccentricity at the beam-‐to-‐column connecKon.

NEd

Nb,z,Rd

+My,Ed

Mb,Rd

+1.5 Mz,Ed

Mc,z,Rd

≤1

The moment components are small for simple construcKon, the interacKon factors can be conservaKvely simplified to :

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Example 1 : Design of column with combined bending and axial load

Thank You

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Topic 8 - steel columns