ADVANCED DESIGN OF STEEL AND COMPOSITE STRUCTURES · PDF fileADVANCED DESIGN OF STEEL AND COMPOSITE STRUCTURES Aldina Santiago ... Buckling curve acc. to EC3-1-1, ... plane flexural

ADVANCED DESIGN OF STEEL AND COMPOSITE STRUCTURES

Aldina SantiagoLecture B.3: 22/2/2017European Erasmus Mundus Master Course

Sustainable Constructions under Natural Hazards and Catastrophic Events

520121-1-2011-1-CZ-ERA MUNDUS-EMMC

Module A – Design of industrial buildings using non-uniform members

European Erasmus Mundus Master Course

Sustainable Constructions under Natural Hazards

and Catastrophic Events

CONTENTS

Module B – Design of industrial buildings using non-uniform members

B.3 – Design of non-uniform members

1 – Introduction2 – Non-uniform members – approaches and problems 4 – Design resistance of non-uniform members (clause 6.3.4)5 – Example

Introduction





Introduction

q Tapered steel members are used in steel structures- Structural efficiency à optimization of cross section capacity à

saving of material- Aesthetical appearance

Multi-sport complex – Coimbra, Portugal Construction site in front of the Central Station,Europaplatz, Graz, Austrial





q Tapered members are commonly used in steel frames:- industrial halls, warehouses, exhibition centers, etc.

- Adequate verification procedures are then required for these types of structures!

Introduction





q However, there are several difficulties in performing the stabilityverification of structures composed of non-uniform members;

Introduction

- Guidelines are inexistent or not clear for the designer.

- Due to this reason, simplifications that are not mechanicallyconsistent are adopted. These may be either too conservative oreven Unconservative!

Non-uniform members

Approachesand

Problems





q Prismatic members – Clauses 6.3.1 to 6.3.3q Developed for prismatic membersq Sinusoidal imperfections

q Ayrton-Perry type equation:Is maximum at mid span:

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 2 4 6 8 10 12

L(x)

δcr

δ'cr

δ''cr

÷øö

çèæ p=dLxsine)x( 00

÷øö

çèæ pµd¢¢=LxsinEI)x(MII

Rk,y

II

Rk M)x(M

NN)x( +=e OK!

Column

Constant Sinusoidal

Non-uniform members – approaches and problems





q Non - prismatic members


- Analytical expressions for the elastic critical load are not readily available;

- The choice of the critical section for the application of the buckling resistance

formulae is not straightforward.





q Non-uniform members – Clauses 6.3.1 to 6.3.3 apply ???

RkNN

Not Constant

NEd


q Cross section utilization due to applied (first order) forces is notconstant anymore

RkNN






-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 2 4 6 8 10 12

L(x)

δcr

δ'cr

δ''cr

÷øö

çèæ p=dLxsine)x( 00

÷øö

çèæ pµd¢¢=LxsinEI)x(MII

Rk,y

II

Rk M)x(M

NN)x( +=e KO!

Column


First yield criteria:

q Ayrton-Perry type equation:Is it maximum at mid span ???






q Position of the critical cross-section – not at mid span- Account for 2nd order effects; iterative procedure, not practical;

q 1st order critical cross section is considered!







q Variation of cross section class

q Definition of an equivalent class for the member

Class 3Class 2 Class 1

My,Ed

NEd <<< Afy






q Non-uniform members – 2nd order analysis with imperfections

q Definition of local imperfections:q Same problem: e0/L calibrated for prismatic members with

sinusoidal imperfections

Buckling curve acc. to EC3-1-1, Table 6.1

Elastic analysis Plastic analysis

e0/L e0/L

a0 1/350 1/300

a 1/300 1/250

b 1/250 1/200

c 1/200 1/150

d 1/150 1/100






q Non-uniform members – 2nd order analysis with imperfections

q Definition of local imperfections?

Auvent de la Gare Routière – Ermont

Barajas Airport, Madrid, Spain

Italy pavilion, World Expo 2010 – Shanghai






q Non-uniform members – GENERAL METHOD (clause 6.3.4)


General method allows the verification of the resistance to lateral and lateral torsional buckling for structural components such as:

– single members, built-up or not, uniform or not, with complex support conditions or not, or

– plane frames or subframes composed of such members,






αult,k

χ χLT

χop = Minimum (χ, χLT)

χop = Interpolated (χ, χLT)

In-plane resistance

αcr,op

Buckling curve

1/ 1, ³Mkultop gac

In-Plane GMNIA calculations LEA calculations

opcrkultop ,, /aal =

Out-of-planeelastic critical load

1

2

3







αult,k

χ χLT



In-plane resistance

αcr,op

Buckling curve

1/ 1, ³Mkultop gac




1

2

3


Minimum load amplifiedof the design loads toreach the characteristicresistance of the mostcritical cross-section ofthe structural component,without lateral-torsionalbuckling, but accountingfor all effects of the in-plane geometricaldeformations andimperfection (global andlocal): NRK / NEd






αult,k

χ χLT



In-plane resistance

αcr,op

Buckling curve

1/ 1, ³Mkultop gac




1

2

3


Minimum amplified for thein plane the design loadsto reach the elastic criticalresistance of thestructural component, withrespect to lateral andlateral-torsional buckling,without accounting for in-plane flexural buckling:Ncr / NEd .FE can be used todetermine αult,k and αcr,op.






αult,k

χ χLT



In-plane resistance

αcr,op

Buckling curve

1/ 1, ³Mkultop gac




1

2

3


Global non-dimensionalslenderness






αult,k

χ χLT



In-plane resistance

αcr,op

Buckling curve

1/ 1, ³Mkultop gac




1

2

3


Reduction factor for:- Flexural buckling- Lateral torsional

bucklingEach calculated for theglobal non-dimensionalslenderness λop.






αult,k

χ χLT



In-plane resistance

αcr,op

Buckling curve

1/ 1, ³Mkultop gac




1

2

3


χop is the reduction factor forthe non-dimensionalslenderness λop, to takeaccount of lateral and lateraltorsional buckling.






αult,k

χ χLT



In-plane resistance

αcr,op

Buckling curve

1/ 1, ³Mkultop gac




1

2

3






Non-uniform members – example

N = cte = 80.0 kN











Verification of the cross-section• The cross-section resistance is checked using clauses 6.2.8 (bending and shear)

and 6.2.9 (bending and axial force) for each class of cross-section

• The utilization ratio a of the cross-section is given by the ration between the normof the applied internal forces and the norm of the bending and axial resistancealong the same load value:

𝛼 =𝑁$%& + 𝑀),$%

&�

𝑁,-.& + 𝑀),,-.&�

≤ 1

𝛼 =𝑁$% + 𝑀),$%

𝐴𝑓𝑦 +𝑊𝑒𝑙𝑦𝑓𝑦≤ 1

Class 1 or 2: being Nmax and My,max the valuesobtained in the interaction curve

Class 3:






Verification of the cross-section

Critical cross-section at x = 4,14 m(Class 1)






General method (cl. 6.3.4 using the properties of the critical cross-section)In-plane buckling resistance

𝛼89:,; =𝑁$%

𝜒)𝑁=; 𝛾?@A+ 𝑘))

𝑀),$%

𝜒C:𝑀),=; 𝛾?@A

D@

=1

0.55 = 1.819

(being 𝜒C: = 1)

Out-of-plane buckling resistanceEquations to calculate 𝛼JK,LM are dificult to apply!

Numerically, 𝛼JK,LM = 1.482

Buckling resistance 𝜆LM =𝛼89:,;𝛼JK,LM

�= 1.108






General method (cl. 6.3.4 using the properties of the critical cross-section)For x = 4.14 m, buckling curves for out-of-plane buckling and lateral-torsional

buckling are curve c.

𝜆LM = O𝑐𝑢𝑟𝑣𝑒𝑐 −> 𝜒V = 0.48𝑐𝑢𝑟𝑣𝑒𝑐 − 𝜒CX = 0.48

𝜒LM = min 𝜒V; 𝜒CX = interp 𝜒V; 𝜒CX = 0.48

So, ,therefore buckling resistance is not verified!𝜒LM𝛼89:,;𝛾?@

=0.48×1.819

1 = 0.873 < 1

The ratio of utilization is 1/0.873 = 1.15 (15% higher than permitted)!






General method (cl. 6.3.4 using the properties of the critical cross-section)

Method Util. ratio Diff (%)GMNIA 0.88

Cla

use

6.3.

3 x = xcr + eq. properties for critical loads 1.22 39

x = 0 1.16 32.8

X = xcr 1.23 40.8

X = L 1.51 72.7

General method (theoretical) - xcr 1.15 30.8





REFERENCES

Simões da Silva, L, Rimões, R and Gervásio, H -“Design of Steel Structures”, Ernst & Sohn and ECCSPress, Brussels, 2010.

• Marques, L., Taras, A., Simões da Silva, L., Greiner, R. and Rebelo, C., “Development of a consistent design procedure for tapered columns", Journal of Constructional Steel Research, 72, pp. 61–74 (2012). http://dx.doi.org/10.1016/j.jcsr.2011.10.008

• Marques, L., Simões da Silva, L., Greiner, R., Rebelo, C. and Taras, A., “Development of a consistent design procedure for lateral-torsional buckling of tapered beams”, Journal of Constructional Steel Research, 89, pp. 213-235 (2013). http://dx.doi.org/10.1016/j.jcsr.2013.07.009





REFERENCES

• Marques, L., Simões da Silva, L., Rebelo, C. and Santiago, A., “Extension of the interaction formulae in EC3-1-1 for the stability verification of tapered beam-columns”, Journal of Constructional Steel Research, 100, pp. 122-135 (2014). http://dx.doi.org/10.1016/j.jcsr.2014.04.024

• Marques, L., Simões da Silva, L. and Rebelo, C., “Rayleigh-Ritz procedure for determination of the critical loads of tapered columns”, Steel and Composite Structures, 16(1), pp. 47-60 (2014). http://dx.doi.org/10.12989/scs.2014.16.1.047

• Simões da Silva, L., Marques, L. and Rebelo, C., (2010) “Numerical validation of the general method in EC3-1-1: lateral, lateral-torsional and bending and axial force interaction of uniform members”, Journal of Constructional Steel Research, 66, pp. 575-590 (2010), http://dx.doi.org/10.1016/j.jcsr.2009.11.003





SOFTWARE

LTBeamN - “Elastic Lateral Torsional Buckling of Beams”, v1.0.1, CTICM, 2013.http://www.steelbizfrance.com/telechargement/desclog.aspx?idrub=1&lng=2SemiComp Member Design – Design resistance of prismatic beam-columns”, Greiner et al, RFCS, 2011.http://www.steelconstruct.com

ECCS Steel Member Calculator – Design ofcolumns, beams and beam-columns to EC3-1-1”, Silva et al, ECCS, 2015. http://www.steelconstruct.com

www.steelconstruct.comwww.cmm.pt





ACKNOWLEDGEMENTS

• This lecture was prepared for the Edition 2 of SUSCOS (2013/15) by LUÍS SIMÕES DA SILVA and LILIANA MARQUES (UC).

• This lecture was improved for the Edition 3 of SUSCOS (2014/16) by LUÍS SIMÕES DA SILVA (UC).

• This lecture was improved for the Edition 4 of SUSCOS (2015/17) by ALDINA SANTIAGO (UC).

The SUSCOS powerpoints are covered by copyright and are for the exclusive use by the SUSCOS teachers in the framework of this Erasmus Mundus Master. They may be improved by the various teachers throughout the different editions.

Documents

ADVANCED DESIGN OF STEEL AND COMPOSITE STRUCTURES · PDF fileADVANCED DESIGN OF STEEL AND COMPOSITE STRUCTURES Aldina Santiago ... Buckling curve acc. to EC3-1-1, ... plane flexural