ADVANCED DESIGN OF STEEL AND COMPOSITE STRUCTURES

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1 ADVANCED DESIGN OF STEEL AND COMPOSITE STRUCTURES Aldina Santiago Lecture B.3: 22/2/2017 European Erasmus Mundus CZ-ERA MUNDUS-EMMC

2 CONTENTS Module B Design of industrial buildings using non-uniform members B.3 Design of non-uniform members 1 Introduction 2 Non-uniform members approaches and problems 4 Design resistance of non-uniform members (clause 6.3.4) 5 Example

3 Introduction

4 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

5 Introduction 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!

6 Introduction q However, there are several difficulties in performing the stability verification of structures composed of non-uniform members; - Guidelines are inexistent or not clear for the designer. - Due to this reason, simplifications that are not mechanically consistent are adopted. These may be either too conservative or even Unconservative!

7 Non-uniform members Approaches and Problems

8 Non-uniform members approaches and problems q Prismatic members Clauses to q Developed for prismatic members q Sinusoidal imperfections d x) = e 0 ( 0 æ px sinç è L ö ø Column δcr δ'cr δ''cr L(x) -0.4 q Ayrton-Perry type equation: Is maximum at mid span: N e( x) = + N Rk M II (x) Constant M M II (x) y,rk = æ px EId µ sinç è L Sinusoidal OK! ö ø

9 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.

10 Non-uniform members approaches and problems q Non-uniform members Clauses to apply??? N Ed q Cross section utilization due to applied (first order) forces is not constant anymore N N Rk Not Constant N N Rk

11 Non-uniform members approaches and problems q Non-uniform members Clauses to apply??? Column δcr δ'cr d x) = e 0 ( 0 æ px sinç è L ö ø 0.6 δ''cr L(x) M II (x) = æ px EId µ sinç è L ö ø q Ayrton-Perry type equation: Is it maximum at mid span??? First yield criteria: N e( x) = + N Rk M M II (x) y,rk KO!

12 Non-uniform members approaches and problems q Non-uniform members Clauses to apply??? q Position of the critical cross-section not at mid span - Account for 2nd order effects; iterative procedure, not practical; q 1 st order critical cross section is considered!

13 Non-uniform members approaches and problems q Non-uniform members Clauses to apply??? q Variation of cross section class Class 3 Class 2 Class 1 My,Ed q Definition of an equivalent class for the member NEd <<< Af y

14 Non-uniform members approaches and problems q Non-uniform members 2 nd order analysis with imperfections q Definition of local imperfections: q Same problem: e 0 /L calibrated for prismatic members with sinusoidal imperfections Buckling curve acc. to Elastic analysis Plastic analysis EC3-1-1, Table 6.1 e 0 /L e 0 /L a 0 1/350 1/300 a 1/300 1/250 b 1/250 1/200 c 1/200 1/150 d 1/150 1/100

Barajas Airport, Madrid, Spain Auvent de

15 Non-uniform members approaches and problems q Non-uniform members 2 nd order analysis with imperfections q Definition of local imperfections? Barajas Airport, Madrid, Spain Auvent de la Gare Routière Ermont Italy pavilion, World Expo 2010 Shanghai

16 Non-uniform members approaches and problems 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,

17 Non-uniform members approaches and problems q Non-uniform members GENERAL METHOD (clause 6.3.4) 1 In-plane resistance In-Plane GMNIA calculations Out-of-plane elastic critical load LEA calculations α ult,k α cr,op 2 l = a op ult k cr, op, /a Buckling curve 3 χ op = Minimum (χ, χ LT ) χ χ LT χ op = Interpolated (χ, χ LT ) c op a ult, k / g M 1 ³ 1

18 Non-uniform members approaches and problems q Non-uniform members GENERAL METHOD (clause 6.3.4) Minimum load amplified of the design loads to reach the characteristic resistance of the most critical cross-section of the structural component, without lateral-torsional buckling, but accounting for all effects of the inplane geometrical deformations and imperfection (global and local): N RK / N Ed 1 In-plane resistance 3 In-Plane GMNIA calculations 2 α ult,k l = χ op = Minimum (χ, χ LT ) Buckling curve χ a χ LT LEA calculations α cr,op op ult k cr, op, /a Out-of-plane elastic critical load χ op = Interpolated (χ, χ LT ) c op a ult, k / g M 1 ³ 1

19 Non-uniform members approaches and problems q Non-uniform members GENERAL METHOD (clause 6.3.4) Minimum amplified for the in plane the design loads to reach the elastic critical resistance of the structural component, with respect to lateral and lateral-torsional buckling, without accounting for inplane flexural buckling: N cr / N Ed. FE can be used to determine α ult,k and α cr,op. 1 In-plane resistance 3 In-Plane GMNIA calculations 2 α ult,k l = χ op = Minimum (χ, χ LT ) Buckling curve χ a χ LT LEA calculations α cr,op op ult k cr, op, /a Out-of-plane elastic critical load χ op = Interpolated (χ, χ LT ) c op a ult, k / g M 1 ³ 1

20 Non-uniform members approaches and problems q Non-uniform members GENERAL METHOD (clause 6.3.4) 1 In-plane resistance In-Plane GMNIA calculations Out-of-plane elastic critical load LEA calculations α ult,k α cr,op Global slenderness non-dimensional 2 l = a op ult k cr, op, /a Buckling curve 3 χ op = Minimum (χ, χ LT ) χ χ LT χ op = Interpolated (χ, χ LT ) c op a ult, k / g M 1 ³ 1

21 Non-uniform members approaches and problems q Non-uniform members GENERAL METHOD (clause 6.3.4) 1 In-plane resistance In-Plane GMNIA calculations Out-of-plane elastic critical load LEA calculations α ult,k α cr,op Reduction factor for: - Flexural buckling - Lateral torsional buckling Each calculated for the global non-dimensional slenderness λ op. 3 2 l = χ op = Minimum (χ, χ LT ) op ult k cr, op Buckling curve χ a, /a χ LT χ op = Interpolated (χ, χ LT ) c op a ult, k / g M 1 ³ 1

22 Non-uniform members approaches and problems q Non-uniform members GENERAL METHOD (clause 6.3.4) 1 In-plane resistance In-Plane GMNIA calculations Out-of-plane elastic critical load LEA calculations α ult,k α cr,op 2 l = a op ult k cr, op, /a Buckling curve χ op is the reduction factor for the non-dimensional slenderness λ op, to take account of lateral and lateral torsional buckling. 3 χ op = Minimum (χ, χ LT ) χ χ LT χ op = Interpolated (χ, χ LT ) c op a ult, k / g M 1 ³ 1

23 Non-uniform members approaches and problems q Non-uniform members GENERAL METHOD (clause 6.3.4) 1 In-plane resistance In-Plane GMNIA calculations Out-of-plane elastic critical load LEA calculations α ult,k α cr,op 2 l = a op ult k cr, op, /a Buckling curve 3 χ op = Minimum (χ, χ LT ) χ χ LT χ op = Interpolated (χ, χ LT ) c op a ult, k / g M 1 ³ 1

24 European Erasmus Mundus Non-uniform members example N = cte = 80.0 kn

25 Non-uniform members example

26 Non-uniform members example Verification of the cross-section The cross-section resistance is checked using clauses (bending and shear) and (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 norm of the applied internal forces and the norm of the bending and axial resistance along the same load value: Class 1 or 2: α = N & & $% + M ),$% & & N,-. + M ),,-. 1 being N max and M y,max the values obtained in the interaction curve Class 3: α = N $% + M ),$% Af y + Wely f y 1

27 Non-uniform members example Verification of the cross-section Critical cross-section at x = 4,14 m (Class 1)

28 Non-uniform members example General method (cl using the properties of the critical cross-section) In-plane buckling resistance α 89:,; = χ ) N =; N $% Aγ?@ M ),$% + k )) χ C: M ),=; A γ?@ D@ = = (being χ C: = 1) Out-of-plane buckling resistance Equations to calculate α JK,LM are dificult to apply! Numerically, α JK,LM = Buckling resistance λ LM = α 89:,; α JK,LM = 1.108

29 Non-uniform members example General method (cl 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. λ LM = O curve c > χ V = 0.48 curve c χ CX = 0.48 χ LM = min χ V ; χ CX = interp χ V ; χ CX = 0.48 So, α 89:,; χ LM = γ?@ 1 = < 1,therefore buckling resistance is not verified! The ratio of utilization is 1/0.873 = 1.15 (15% higher than permitted)!

30 Non-uniform members example General method (cl using the properties of the critical cross-section) Method Util. ratio Diff (%) GMNIA 0.88 Clause x = x cr + eq. properties for critical loads x = X = x cr X = L General method (theoretical) - x cr

REFERENCES Simões da Silva, L, Rimões, R and Gervásio, H - Design of Steel Structures, Ernst & Sohn and ECCS

, Development of a consistent design procedure for tapered columns", Journal of Constructional Steel

31 REFERENCES Simões da Silva, L, Rimões, R and Gervásio, H - Design of Steel Structures, Ernst & Sohn and ECCS Press, Brussels, 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 (2012). 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 (2013).

32 REFERENCES 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 (2014). 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 (2014). 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 (2010),

SOFTWARE LTBeamN - Elastic Lateral Torsional Buckling of Beams, v1.0.1, CTICM, 2013. http://www.steelbizfrance.com/telechargement/desclog. aspx?

33 SOFTWARE LTBeamN - Elastic Lateral Torsional Buckling of Beams, v1.0.1, CTICM, aspx?idrub=1&lng=2 SemiComp Member Design Design resistance of prismatic beam-columns, Greiner et al, RFCS, ECCS Steel Member Calculator Design of columns, beams and beam-columns to EC3-1-1, Silva et al, ECCS,

34 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.

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