ADVANCED DESIGN OF STEEL AND COMPOSITE STRUCTURES

ADVANCED DESIGN OF STEEL AND COMPOSITE STRUCTURES

Luís Simões da Silva Lecture 1: 20/2/2014

European Erasmus Mundus Master Course Sustainable Constructions

under Natural Hazards and Catastrophic Events 520121-1-2011-1-CZ-ERA MUNDUS-EMMC

Module A – Design of non-uniform members

European Erasmus Mundus Master Course

Sustainable Constructions under Natural Hazards

and Catastrophic Events

CONTENTS

Module A – Design of non-uniform members 1 – Introduction 2 – Non-uniform members – approaches and problems 3 – Tapered columns

3.1 – Differential equation 3.2 – Elastic critical load 3.3 – Ayrton-Perry formulation

4 – Design resistance of tapered columns and beams 4.1 – Derivation 4.2 – Tapered columns 4.2.1 – Design methodology 4.2.2 – Example 4.3 – Tapered beams 4.3.1 – Elastic critical moment 4.3.2 – Design methodology 4.3.3 – Example

5 – Beam-columns

Introduction





Introduction

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

saving of material q  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: q  industrial halls, warehouses, exhibition centers, etc.

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

Introduction





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

q  Guidelines are inexistent or not clear for the designer

q  Due to this reason simplifications that are not mechanically consistent are adopted

q  These may be either too conservative or even q  Unconservative!

Introduction

Non-uniform members

Approaches and

Problems





q  Prismatic members – Clauses 6.3.1 to 6.3.3 q  Developed for prismatic members q  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

⎟⎠⎞⎜

⎝⎛ π=δLxsine)x( 00

⎟⎠⎞⎜

⎝⎛ π∝δ′′=LxsinEI)x(MII

Rk,y

II

Rk M)x(M

NN)x( +=ε OK!

Column

Constant Sinusoidal

Non-uniform members – approaches and problems





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

q  Cross section utilization due to applied (first order) forces is not constant anymore.

RkNN

Not Constant

NEd







q  Ayrton-Perry type equation: Is it 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

⎟⎠⎞⎜

⎝⎛ π=δLxsine)x( 00

⎟⎠⎞⎜

⎝⎛ π∝δ′′=LxsinEI)x(MII

Rk,y

II

Rk M)x(M

NN)x( +=ε KO!

Column







q  Position of the critical cross-section – not at mid span q  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)

αult,k

χ χLT

χop = Minimum (χ, χLT)

χop = Interpolated (χ, χLT)

In-plane resistance

αcr,op

Buckling curve

1/ 1, ≥Mkultop γαχ

In-Plane GMNIA calculations LEA calculations

opcrkultop ,, /ααλ =

Out-of-plane elastic critical load

1

2

3







q  αult,k should account for local second order effects?

q  If so: again à problem with definition of imperfections q  Prismatic column – 6.3.4 vs. 6.3.1

q  Higher in-plane second order effects lead to a decrease in the out-of-plane reduction factor

q  Even for the case of beam-columns this effect is not as restrictive.

1

80

85

90

95

100

105

0 0.5 1 1.5 2 2.5 3

% 6

.3.4

/ 6.3

.1

λ_z

Φ=∞ Rolled

IPE 360IPE 200HE 550 BHE 500 AHE 450 BHE 400 BHE 360 BHE 340 BHE 300 BHE 200 A







q  Definition of a proper buckling curve

q  For some cases existing buckling curves may even be unconservative!

Curve b

Curve d

Curve c

Cu

Curve a

2







q  Reduction factor – minimum or interpolation q  Minimum

q  May be too conservative q  Does not follow buckling mode correctly

q  Interpolation q  Provides a transition between FB (My=0) and LTB (N=0) q  What type of function for a correct mode transition?

3






q  Non-uniform members – FEM numerical analysis

q  Problem with definition of imperfections q  Requires a high experience in FEM modeling from the user in

order to achieve reliable results q  Limited guidelines

q  For the most simple cases it is preferable to provide simple rules which include as much as possible the real behavior of the member


Tapered columns





q  Equilibrium

n(x)

Nconc

x

x

δ

dxdxdMM +

dxdxdNN +

M

N

Q

dx

dxdxdQQ +

dδ

A

B

n(x)

ξ

Bδ

dx

dδ

n(x)

A

η

∫+=L

xconc dnNxN ξξ )()(

Tapered columns Differential equation





q  Equilibrium

q  Eq. Moments in B

q  Eq. Horizontal

x

δ

dxdxdMM +

dxdxdNN +

M

N

Q

dx

dxdxdQQ +

dδ

A

B

n(x)

∫+=L

xconc dnNxN ξξ )()(

dxdyxN

dxdMQ

dnMdxdxdMMQdxdyxN

dx

)(

0

0

)().(0

−=→

=

≈

−+⎟⎠⎞⎜

⎝⎛ +−+ ∫ !"!#$

ξηξ

0=→+=dxdQdx

dxdQQQ

+ neglect 2nd order terms

⎟⎠⎞⎜

⎝⎛−==

dxdxN

dxd

dxMd

dxdQ δ)(0

2

2






q  Equilibrium

x

δ

dxdxdMM +

dxdxdNN +

M

N

Q

dx

dxdxdQQ +

dδ

A

B

n(x)

⎟⎠⎞⎜

⎝⎛−=

dxdxN

dxd

dxMd δ)(0 2

2

2

2)()(dxdxEIxM δ−= +

0)()(2

2

2

2=⎟

⎠⎞⎜

⎝⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛dxdxN

dxd

dxdxI

dxdE δδ

( ) ( ) 0)()( =′′⋅+″′′⋅ δδ xNxIE⎪⎩

⎪⎨

⎧

===

)()()()()()(

xxxnxnxNxN

cr

Edcr

Edcr

δδαα






q  Differential equation

q  Proposal for determination of major axis critical load of tapered I

columns (Marques et al, 2014)

( )( )min.ymax.yI

I156.0

Imin,crTap,cr

II1tan04.01ANAN

=

−⋅−=→⋅= −

γγγ

Tapered columns Elastic critical load

( ) ( ) 0)()( =′′⋅+″′′⋅ δδ xNxIE





q  Equilibrium

q  First yield criterion

( ) ( ) 0)()( =′′⋅+″′′⋅ δδ xNxIE⎪⎩

⎪⎨⎧

===

)()(0)(

)()(

xxxn

xNxN

cr

Edcr

δδ

α

( ) ( ) 0)()()( 0 =′′+′+″′′ δδδ xNxNxEI⎪⎩

⎪⎨⎧

−=

=

)()(

)()(

0 xx

xNxN

bcr

b

Edb

δαα

αδ

α

⎥⎦

⎤⎢⎣

⎡ ′′−

−=′′−= )()()()()( 0 xxEIxxEIxMbcr

b δαα

αδ

)(

))(()(

)()(

)()(

)()()(

0

xM

xxEI

xNxN

xMxM

xNxNx

R

bcr

b

R

Edb

RR

Edb⎥⎦

⎤⎢⎣

⎡ ′′−−

+=+=δ

ααα

ααε

Tapered columns Ayrton-Perry formulation





⎥⎦

⎤⎢⎣

⎡ ′′−⎥⎦

⎤⎢⎣

⎡

−+=

)(.))().((

)()(

)()(1

)()(1 02 IIcEdcr

IIccr

IIc

IIcR

IIcR

IIc

IIc

IIcII

c xNxxEI

xMxNe

xx

xxα

δχλ

χχ

q  Assumption for imperfection

q  Introducing slenderness and reduction factor definitions

q  At the critical location ε(xcII)=1

00 )()( exx crδδ =

( )

)(

)()1()(

)()(

)()(

)()(

)(

0

0

xM

exxEI

xNxN

xMxM

xNxN

xR

crbcr

b

R

Edb

RR

Edb ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ ′′′′−⋅

−+=+=

− !!"!!#$ δδ

ααα

ααε

cr

EdR xNxNx

αλ )(/)()( =

)(/)()(

xNxNx

EdR

bαχ =

( )⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡ ′′−⎥⎦

⎤⎢⎣

⎡

−⋅+=

)()(

)()(

)()()(

)()(

1

1)()()( 0 xMxM

xNxN

xNxxEI

xMxN

exxxR

cR

cR

R

crEd

cr

cR

cR

cr

b αδ

ααχχε

( )2.0)(3 −IIcEC xλα

Tapered columns Ayrton-Perry formulation

Design resistance of tapered columns and beams





q  Boundary conditions: Fork supports End cross sections remain straight

L/1000

q  Member imperfections: With amplitude e0=L/1000 Same shape as the buckling mode

q  Material imperfections: q  Calculations:

LBA GMNIA

f y

E= 210 GPaf y= 236 MPa

εy ε

q  Material: Perfect elastic-plastic

0.3 f y

0.3 f y

0.3 f y

0.5 f y

0.5 f y

0.5 f y

0.8h

0.2b

fy0.25 fy






q  Columns

q  Beams

( )⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡ ′′−⎥⎦

⎤⎢⎣

⎡

−⋅+=

)()(

)()(

)()()(

)()(

1

1)()()( 0 xMxM

xNxN

xNxxEI

xMxN

exxxR

cR

cR

R

crEd

cr

cR

cR

cr

b αδ

ααχχε

Rk,Rk,

II

Rk,z

z

Rk,z

IIz

MEI

MM

MvEI

MM

ϖ

ω

ϖ

ϖ φ′′−=

′′−=

RkRk

II

MEI

MM δ ′′−=

RkNN

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡

λ

λ

λ

λ⎥⎦

⎤⎢⎣

⎡×

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+

+δ′′−ξ

×⎥⎥⎦

⎤

⎢⎢⎣

⎡

λ

λ⎥⎦

⎤⎢⎣

⎡

χλ−

χ+χ=ε)x(

)x(

)x(

)x()x(A)x(W

)x(W)x(A

2h

NM

1

2)x(h

MN

1

N)x(EI)x(

)x(

)x()x(W)x(Ae

)x()x(1

)x()x()x(IIc

2LT

IIc

2z

2z

2LT

IIc

IIcz

zmin

Tap,z,cr

Tap,cr

Tap,cr

Tap,z,cr

Tap,z,cr

zminh,cr

IIc

2z

IIc

2LT

IIcz

IIc

0

LT

2LT

LTLT

Rk,y

y

MM






q  Columns

q  Beams

( )⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡ ′′−⎥⎦

⎤⎢⎣

⎡

−⋅+=

)()(

)()(

)()()(

)()(

1

1)()()( 0 xMxM

xNxN

xNxxEI

xMxN

exxxR

cR

cR

R

crEd

cr

cR

cR

cr

b αδ

ααχχε

( )2.0)x( IIc3EC −λα

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡

λ

λ

λ

λ⎥⎦

⎤⎢⎣

⎡×

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+

+δ′′−ξ

×⎥⎥⎦

⎤

⎢⎢⎣

⎡

λ

λ⎥⎦

⎤⎢⎣

⎡

χλ−

χ+χ=ε)x(

)x(

)x(

)x()x(A)x(W

)x(W)x(A

2h

NM

1

2)x(h

MN

1

N)x(EI)x(

)x(

)x()x(W)x(Ae

)x()x(1

)x()x()x(IIc

2LT

IIc

2z

2z

2LT

IIc

IIcz

zmin

Tap,z,cr

Tap,cr

Tap,cr

Tap,z,cr

Tap,z,cr

zminh,cr

IIc

2z

IIc

2LT

IIcz

IIc

0

LT

2LT

LTLT

( )2.0)x( IIczLT −λα

)(

)(IIcuniformnon

IIcuniform

x

x

−η

η






q  Columns

q  Beams

( )⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡ ′′−⎥⎦

⎤⎢⎣

⎡

−⋅+=

)()(

)()(

)()()(

)()(

1

1)()()( 0 xMxM

xNxN

xNxxEI

xMxN

exxxR

cR

cR

R

crEd

cr

cR

cR

cr

b αδ

ααχχε

( ) )x(2.0)x()x()x()x(1

1)x()x(1 IIc

IIc

IIc3ECII

cIIc

2IIc

IIc β×−λα

χλ−χ+χ=

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡

λ

λ

λ

λ⎥⎦

⎤⎢⎣

⎡×

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+

+δ′′−ξ

×⎥⎥⎦

⎤

⎢⎢⎣

⎡

λ

λ⎥⎦

⎤⎢⎣

⎡

χλ−

χ+χ=ε)x(

)x(

)x(

)x()x(A)x(W

)x(W)x(A

2h

NM

1

2)x(h

MN

1

N)x(EI)x(

)x(

)x()x(W)x(Ae

)x()x(1

)x()x()x(IIc

2LT

IIc

2z

2z

2LT

IIc

IIcz

zmin

Tap,z,cr

Tap,cr

Tap,cr

Tap,z,cr

Tap,z,cr

zminh,cr

IIc

2z

IIc

2LT

IIcz

IIc

0

LT

2LT

LTLT

( )( ) )x()x(

)x(2.0)x()x()x(1

)x()x(11)x( IIcII

c2z

IIc

2LTII

czLTIIcLT

IIc

2LT

IIcLTII

cLTc β×⎥⎥⎦

⎤

⎢⎢⎣

⎡

λ

λ−λα×χλ−

χ+χ=→=ε






q  Interpretation of xcII and β – example (column)

N N

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 0.5 1 1.5 2 2.5 3 3.5

xcII/L

λy(xcI)

1.00

1.17

1.33

1.50

2.00

3.00

5.00

γh=

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3 3.5

β(xcII)

λy(xcII)

1.00

1.50

2.00

2.50

3.00

4.00

5.00

γh=

xc






q  Simplification of the analytical models

q  Columns

q  Beams

λ

IIcx;β II

cx lim,lim ;β

λ

χ

Icx;0=β

( )( ) !"#lim

IIlim,c

2z

IIlim,c

2LTII

lim,czLTIIcLT

IIc

2LT

IIlim,cLTII

lim,cLT 1)x(

)x(2.0)x(

)x()x(1

)x()x(1

β×

⎥⎥⎦

⎤

⎢⎢⎣

⎡

λ

λ−λα×

χλ−

χ+χ=

( ) !lim

12.0)x()x()x(1

1)x()x(1 IIlim,cII

lim,cIIlim,c

2IIlim,c

IIlim,c

β×−λα

χλ−χ+χ=

1






q  Simplification of the analytical models

q  Columns

q  Beams

q  Transformation of variables

λ

IIcx;β II

cx lim,lim ;β

λ

χ

Icx;0=β

( )( ) !"#lim

IIlim,c

2z

IIlim,c

2LTII

lim,czLTIIcLT

IIc

2LT

IIlim,cLTII

lim,cLT 1)x(

)x(2.0)x(

)x()x(1

)x()x(1

β×

⎥⎥⎦

⎤

⎢⎢⎣

⎡

λ

λ−λα×

χλ−

χ+χ=

( ) !lim

12.0)x()x()x(1

1)x()x(1 IIlim,cII

lim,cIIlim,c

2IIlim,c

IIlim,c

β×−λα

χλ−χ+χ=

)x()x( Ic

IIlim,c λ×ϕ=λ ϕχ=χ /)x()x( I

cIIlim,c

)x()x(

Ick,ult

IIlim,ck,ult

αα

=ϕ

1

2






1.  Required data

Calculate ϕ

(…) Calculate αcr

Calculate utilization ε=NEd/NRk based on Semi-Comp approach

)x(N)x(N)x( IcEd

IcRkI

ck,ilt =α

xIc

COLUMNS – DESIGN METHODOLOGY






2.  Application of the method


crIck,ult

Ic )x()x( αα=λ( )2.0)x( I

c −λ×ϕ×α=η

1)x(

)x(Ic

22

Ic ≤

λ×ϕ−φ+φ

ϕ=χ

( ))x()x(15.0 Ic

2Ic

2λ×ϕ+λ×η×ϕ+×=φ

1)x()x( bIck,ult

Ic ≥α=α×χ

NEd αb x NEd

3.  Verification






q  Necessary parameters

q  Critical load multiplier – αcr q  May be numerical q  From the literature q  For in-plane loading, a formula

was developed considering Rayleigh-Ritz Method :


( )( )min.max.

156.0min,, 1tan04.01

yyI

IIcrTapcr

IIANAN

=

−⋅−=→⋅= −

γγγ

-15

-10

-5

0

5

10

15

20

0 10 20 30 40

Diff (%)

γI

100x10 H&CHEB300 H&CIPE200 H&C100x10 EQUHEB300 EQUIPE200 EQU100x10 L&al.HEB300 L&al.IPE200 L&al.







q  Imperfection factor αy or αz


FB out-of-plane FB in-plane

Hot-rolled: Welded: Hot-rolled: Welded:

α 0.49 0.64 0.34 0.45

η - ≤0.34 - ≤0.27

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.0 0.5 1.0 1.5

χy

λ̅y

α=0.34 (EC3)α=0.45 (Best fit)α=0.45 + η≤0.27Euler







q  Overstrength factor φy or φz


FB out-of-plane FB in-plane

⎥⎦

⎤⎢⎣

⎡ −++

h

hhw

Aht

γγγ

10)1)(41(

1min

111

min

min

+−+

h

hw

Ath

γγ






q  Results q  Out-of-plane

(γh=hmax/hmin=1.8)


0.0

0.2

0.4

0.6

0.8

1.0

0 0.5 1 1.5 2

αb

λz(xcI)

EC3-cEulerGMNIAProposal

0.0

0.2

0.4

0.6

0.8

1.0

0 0.5 1 1.5 2 2.5

αb

λy(xcI)

EC3-bEulerGMNIAProposal

q  In-plane (γh=hmax/hmin=6)






q  Possible problems q  Web buckling – critical location varies

q  φ was calibrated considering critical location without local buckling effects!

q  May lead to unsafe results!







q  Flange thickness in vertical plan

COLUMNS - EXAMPLE

α=2.519º

mmtt ff 02.25)cos(/' == α

NEd

L=5m hw,min = 170 mm hw,max = 610 mm b = 206 mm tf = 25 mm tw = 15 mm S235 NEd = 1100 kN L = 5 m

hmin = 220.05 mm hmax = 660.05 mm

γh= 660.05/220.05=3

cw/tw<33εàClass1 <38εà Class2

<42ε àClass4

Flange class: cf/tf=(206/2-15/2)/25=3.82 <9ε à Class1

x=3.65m x=4.55m






q  Minor axis verification

q  Calculation of slenderness at x=xcI

q  Overstrength-factor, φ

q  Determination of imperfection, η

COLUMNS - EXAMPLE hw,min = 170 mm hw,max = 610 mm b = 206 mm tf = 25 mm tw = 15 mm S235 NEd = 1100 kN L = 5 m

222.1310

)13)(341(12850

1505.220110

)1)(41(1min

min =⎥⎦⎤

⎢⎣⎡

×−×+×+=⎥

⎦

⎤⎢⎣

⎡ −++=h

hhwz A

thγγγϕ

( ) ( ) 34.034.0580.02.01222.164.02.0)( =→>=−×=−= ηλϕαη Icx

11100/1.30221100/1.3022

//)(/)()(

min,

==≈=Edhcr

EdIcRk

cr

EdIcRkI

c NNNxNNxNx

αλ

2min

2

min, LEIN h

hcrπ=






q  Minor axis verification

q  Reduction factor

q  Verification

q  Numerical analysis, GMNIA


kNkNxNxN IcPl

IcRdzb 11009.19151.3022634.0)()(,, >=×== χ

1634.01222.1281.1281.1

222.1

)()(

2222≤=

×−+=

×−+=

Ic

Ic

xx

λϕφφ

ϕχ

( ) ( ) 281.11222.134.015.0)(15.0 22=×++=++= I

cxλϕηφ

diffkNN Rdzb %5.068.1904,, →=






q  Major axis verification

q  Critical load, Ncr,y,Tap

q  Numerical buckling analysis, LBA:

COLUMNS - EXAMPLE hw,min = 170 mm hw,max = 610 mm b = 206 mm tf = 25 mm tw = 15 mm S235 NEd = 1100 kN L = 5 m ( )( )

kNkNNANA

cmcm

II

crTapycr

II

h

hI

56.338048681894.3894.31tan04.01

64.1210471132365

min,,,

156.0

2

2

min

max

=×=⋅==−⋅−=

===

− γγ

γ

kNN Tapycr 32516,, = à diff. 3.9%








q  Overstrength-factor, φ



128.11313

128501505.2201

111 2

min

min =+−×+=

+−+=

mmmmmm

Ath

h

hwy γ

γϕ

( ) ( ) 053.027.0053.02.0299.0128.145.02.0)( =→<=−×=−= ηλϕαη Icx

299.01100/56.338041100/1.3022

//)(/)()(

,,

==≈=EdTapycr

EdIcRk

cr

EdIcRkI

c NNNxNNxNx

αλ








q  Verification



kNkNxNxN IcPl

IcRdyb 11001.30221.30221)()(,, >=×== χ

1)(107.1299.0128.1577.0577.0

128.1

)()(

2222=>=

×−+=

×−+= I

cIc

Ic x

xx χ

λϕφφ

ϕχ

( ) ( ) 577.0299.0128.1053.015.0)(15.0 22=×++=++= I

cxλϕηφ

diffkNN Rdzb %01.3022,, →=






1.  Necessary data

Calculate ϕ

(…)

Calculate αcr

BEAMS – DESIGN METHODOLOGY

Calculate utilization ε=MEd/MRk based on Semi-Comp approach

)()(

)(, IcEd

IcRkI

ckilt xMxMx =α

xIc

Calculate xc,limII

(…)






2.  Application of the method


3.  Verification

crIckult

IcLT xx ααλ )()( ,=

2

2 )(LxEINIIcz

crπ≈ )( II

cRk xN

)( IIcz xλ)(TarasLTα

( )2.0)( −×= IczLT xλαη

1)(

)(22

≤×−+

=IcLTLTLT

IcLT

xx

λϕφφ

ϕχ

⎟⎟⎠

⎞⎜⎜⎝

⎛×+××+×= )(

)(

)(15.02

2

2IcLT

IIcz

IcLT

LT xx

x λϕλληϕφ

)(

)(

,

,

IIcelz

IIcely

xW

xW

1)()( , ≥=× bIckult

IcLT xx ααχ

MEd

Ψ MEd

αb x MEd







q  Critical load multiplier – αcr q  May be numerical q  From the literature

q  Imperfection factor αLT

q  Model consistent with recently developed proposals for prismatic beams


Hot-rolled: Welded:

α 49.0)(

)(16.0

lim,,

lim,, ≤IIcelz

IIcely

xW

xW

64.0

)(

)(21.0

lim,,

lim,, ≤IIcelz

IIcely

xW

xW

η - ( )35.023.012.0)x(W)x(W 2

IIlim,cel,z

IIlim,cel,y +ψ−ψ≤







q  xc,limII


For ψ ( ) ( )( )( ) ( )103.012.0/,1214.110

0106.0006.0025.007.018.075.0

lim,

22

−−=−+≥<

≥−−−+−−

hIIchw

h

LxandIf γγγψψγψψψψ

For UDL ( ) ( ) 5.0103.010035.05.0 22 ≤−−−+ hh γγ

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-1 -0.5 0 0.5 1

xc,limII/L

ψ

γh=1γh=1.5γh=2γh=2.6γh=3.1γh=3.6γh=4







q  Overstrength factor φLT


0.9

1.1

1.3

1.5

1.7

1.9

2.1

2.3

-1 -0.5 0 0.5 1

φ

ψ

γw= 1γw= 1.5γw= 2γw= 2.5γw= 3γw= 3.5γw= 4γw= 4.5γw= 5γw= 5.5γw= 6γw= 6.5

0

0.2

0.4

0.6

0.8

1

1.2

0 2 4 6

|ψlim|

γw-1

|ψlim|







q  Overstrength factor φLT


aγ ( ) ( ) ( ) ( )178.01077.01009.010005.0 234 −⋅+−⋅−−⋅+−⋅− wwww γγγγ

ψlim 3232 330114012312106001201 γγγγγγ aaaaaa ⋅+⋅+⋅+⋅−⋅+⋅+

φLT limψψ −< limlim ψψψ ≤≤− limψψ >

A 09.12.19.236.5

973.2718.00665.0

23

456

−⋅−⋅−⋅+

⋅−⋅+⋅−

γγγ

γγγ

aaa

aaa

22.11

12070510581

140080501209037.1132

32

+⋅−⋅+⋅−

⋅+⋅−⋅+−

γγγ

γγγ

aaa

aaa

157.008.0008.0

2−⋅−⋅ γγ aa

B 218.224.527.9

287.53185.11244.0

23

456

−⋅−⋅−⋅+

⋅−⋅+⋅−

γγγ

γγγ

aaa

aaa

0.10.505.10.932

0.4250.1330.02

23

456

−⋅−⋅+⋅−

⋅+⋅−⋅+

γγγ

γγγ

aaa

aaa

0.370.48

0.040.03323

+⋅+

⋅+⋅−

γ

γγ

a

aa

C 0.302.0355.2911.3

314.20.60030.0579

23

456

+⋅+⋅−⋅+

⋅−⋅+⋅−

γγγ

γγγ

aaa

aaa

25.114.002.0

2+⋅−⋅ γγ aa 0.80.06

0.0920.03223

+⋅+

⋅−⋅

γ

γγ

a

aa

UDL:

Ψ: 12 ≥+⋅+⋅ CBA ψψ

05.1015.00025.0 2 ++− γγ aa






q  Results

q  Possible problems: q  Web buckling


q  Influence of shear!

00.20.40.60.8

1

0.2 0.6 1 1.4 1.8

χLT (xcI)

λ̄LT (xcI)

ψ= -1 γh= 3

EulerModel φ= 1.352GMNIA

0

0.2

0.4

0.6

0.8

1

0.2 0.6 1 1.4 1.8

χLT (xcI)

λ̄LT (xcI)

UDL γh= 2.5

Shear interactionEulerModel φ= 1.067GMNIA






q  Flange thickness in vertical plan

q  VEd=300/5 kN = 60 kN Vpl,Rd,min=170x15x10-6 fy=599.25 kN > 2VEd hw,max/tw=610/15=41<72ε

BEAMS - EXAMPLE

α=2.519º

mmtt ff 02.25)cos(/' == α

My,Ed

L=5m hw,min = 170 mm hw,max = 610 mm b = 206 mm tf = 25 mm tw = 15 mm S235 My,Ed = 300 kNm L = 5 m

hmin = 220.05 mm hmax = 660.05 mm

γh= 660.05/220.05=3

cw/tw<124εàClass1

Flange class: cf/tf=(206/2-15/2)/25=3.82 <9ε à Class1






q  Elastic critical moment, numerical analysis

q  x=xcI=5m

BEAMS - EXAMPLE hw,min = 170 mm hw,max = 610 mm b = 206 mm tf = 25 mm tw = 15 mm S235 My,Ed = 300 kNm L = 5 m

kNmM Tapcr 85.2044, =

)()(

)(,

,

xMxM

xRdy

EdyM =ε

0

0.05

0.1

0.15

0.2

0.25

0.3

0 1 2 3 4 5

Série1

238.0)( =LMε







q  Second order failure location, x=xc,limII


733.0300/85.2044300/19.1097)(

)( , ===cr

IckultI

c

xx

αα

λ

03...

214.47.9517.4010

3

3

min,,

max,,

===

===

ψγ

γ

h

ely

elyw mm

mmWW

( ) ( )( ) 063.0106.0006.0025.007.018.075.0/ 22lim, ≥=−−−+−−= hIIc Lx γψψψψ






q  Over-strength factor, φ



03...

214.47.9517.4010

3

3

min,,

max,,

===

===

ψγ

γ

h

ely

elyw mm

mmWW

( ) ( ) ( ) ( )

⎪⎩

⎪⎨⎧

==−==−==

→∀≤=

=−⋅+−⋅−−⋅+−⋅−=

053.1...503.0...097.1...

0

957.1178.01077.01009.010005.0

limlim

234

CBA

a wwww

ψψψ

γγγγγ

1053.12 ≥=+⋅+⋅= CBA ψψϕ

( ) ( )

64.0)()(

21.0

35.023.012.0)()(

2.0)(

lim,,

lim,,

2

lim,,

lim,,

≤=

+−≤−=

IIcelz

IIcely

LT

IIcelz

IIcelyII

cLTLTLT

xWxW

xWxW

x

α

ψψλαη

978.0557.0

150.1)()(

)(

587.0

22

<=

=⋅

=

=

LT

IIc

yIIcII

cLT

LT

LxEIfxA

x

ηπ

λ

α







q  Verification


kNmxMkNxMxM IcEdy

IcRdy

IcRdb 300)(45.82519.1097751.0)()( ,,, =>=×== χ


901.0)()(

)(15.02

lim,

2

2

=⎟⎟

⎠

⎞⎜⎜

⎝

⎛×+××+×= I

cLTIIcz

IcLT

LT xx

x λϕλληϕφ

1752.0)(

)(22

≤=×−+

=IcLTLTLT

IcLT

xx

λϕφφϕχ

diffkNM Rdb %3.395.853, →=


Beam-columns





q  How to solve the problem

VEd

HEd

2nd order effects P-‐Δ+

Global imperfectionsφ e0,y e0,z

0 YES YES YES YES Max. Load factor (≡ GMMIA) -‐1 YES YES YES NO Cross section -‐2a YES YES NO NO Out-‐of-‐plane Member stability procedures L2b YES NO NO NO Member stability procedures Global Lcr,z3 NO NO NO NO Member stability procedures Global Lcr,z Lcr,y

Buckling lengthCheckLocal imperfections

+2nd order effects P-‐δ

Material nonlinearity

Beam-columns







0 YES YES YES YES Max. Load factor (≡ GMMIA) -‐1 YES YES YES NO Cross section -‐2a YES YES NO NO Out-‐of-‐plane Member stability procedures L2b YES NO NO NO Member stability procedures L3 NO NO NO NO Member stability procedures Global Lcr,z Lcr,y




q  Difficulties for each approach

? ? ? 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

Need to calibrate e0/L acc. to new imperfection factors for welded members!

Beam-columns












Calibrate e0/L for new imperfection factors Need to develop

àIn-plane àOut-of-plane

Approaches: àInteraction àGeneralized

slenderness

Beam-columns












q Focus on approach 2a:

àDevelop out-of-plane verification procedure; à To account for LTB in the in-plane stability reduce Mpl by χLT in the cross section check

Beam-columns





q  Adaptation of the Interaction formula

q  kzy factor determined from Method 2 (Annex B) q  Because yy and zz modes are clearly separated

q  Cmy factors determined with MEMBER UTILIZATON (My/My,Rk)

0.1)()(

)()()()(

1,,,

,,

1,,

, ≤+M

IMcRky

IMcLT

IMcEdy

zyM

INcRk

INcz

INcEd

xMxxM

kxNxxN

γχγχ

-1

-0.5

0

0.5

1

0 0.2 0.4 0.6 0.8 1

εM

x/L

ε(Ms)=0.197

ε(Mh)=-1

ψε.ε(Mh)=0.435

( )4.04.0

3.08.011.0435.0917.0

=→<=−−=

−=−=

mimi

smi

s

CCC αψ

ψα

ε

ε

Beam-columns





q  Adaptation of the Interaction formula q  Results

0

10

20

30

40

50

60

70

0 100 200 300 400NEd [kN]

My,Ed [kNm]Interaction

GMNIA

0

20

40

60

80

100

120

0 100 200 300NEd [kN]

My,Ed [kNm]

Interaction

GMNIA

cross section

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

χovMethod

χovGMNIA

Interaction

Low slenderness:

Beam-columns





q  Adaptation of the Interaction formula q  In-plane failure mode

0

20

40

60

80

100

120

0 200 400 600 800NEd [kN]

My,Ed [kNm] CsectionGMNIA λy (xc,I)=0.18GMNIA λy (xc,I)=0.62GMNIA λy (xc,I)=0926.61

0100200300400500600700800

0 1000 2000 3000 4000NEd [kN]

My,Ed [kNm] cross sectionGMNIA λy (xc,I)=0.28GMNIA λy (xc,I)=0.7GMNIA λy (xc,I)=1.396.61

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

χMethod

χGMNIA

6.61

+15%

-5%

Also up to 20% conservative

Beam-columns





Ly≡ L

Lz≡ L/2

Ly / 1000

Lz / 1000 0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

χMethod

χGMNIA

6.61 or 6.62+20%-6%

Also up to 20% conservative

q  Adaptation of the Interaction formula q  In-plane and out-of-plane failure

mode

Beam-columns





q  Member class

BEAM-COLUMNS - EXAMPLE

α=2.519º My,Ed

L=5m hw,min = 170 mm hw,max = 610 mm b = 206 mm tf = 25 mm tw = 15 mm S235 NEd = 1100 kN My,Ed = 300 kNm L = 5 m

NEd

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.2 0.4 0.6 0.8 1

x/L

Class 1Class 2Class 3

ε

Beam-columns





q  Interaction formulae

q  Data:


0.1)()(

)()()(

)(

1,,,

,,

1,,

, ≤+M

IMcRky

IMcLT

IMcEdy

yyM

INcRk

INcy

INcEd

xMxxM

kxNx

xNγχγχ

0.1)()(

)()()()(

1,,,

,,

1,,

, ≤+M

IMcRky

IMcLT

IMcEdy

zyM

INcRk

INcz

INcEd

xMxxM

kxNxxN

γχγχ

1

53.942)5()(

1.3022)0()(

300)(

1100)(

1

,,,

,

,,

,

=

==

==

=

=

M

elyIMcRky

RkINcRk

IMcEdy

INcEd

kNmMxM

kNNxN

kNmxM

kNxN

γ?

?

752.0)5()(

1)0()(

634.0)0()(

,

,

,

=

=

==

==

==

zy

yy

yIMcLT

yINcy

zINcz

kk

x

x

x

χχ

χχ

χχ

Beam-columns





q  Cmy, CmLT


( ) ( ) 942.01.3022

1100299.0128.16.0881.06.06.0

1)()(

)(

6.0

)(11,,

,, =⎟

⎟

⎠

⎞⎜⎜

⎝

⎛×××

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛×

≤

+=

≤

+= !!! "!!! #$!! "!! #$ MINcRk

INcy

INcEdI

Ncyymyyy xNx

xNxCk

γχλϕ

0

0.2360.277

0

0.05

0.1

0.15

0.2

0.25

0.3

0 1 2 3 4 5

881.08.02.08507.0277.0/236.0)(/)(

0

=×+==⇒===

=

smLTmy

helsels

el

CCMM

αεεα

ψ

955.0634.01100

1.302225.0881.0

05.0

1222.105.01

)()(

)(

25.0

05.0

)(05.01

11,

1,

1,

,

,=

×−

≤

×−=

−

≤

−=

!"!#$!! "!! #$

MNcRkNcz

NcEd

LTm

INcz

xNx

xN

C

xzzyk

γχ

λϕ

Beam-columns





q  Verification


0.1701.0

0.1)()(

)()()()(

1,,,

,,

1,,

,

≤

≤+M

IMcRky

IMcLT

IMcEdy

yyM

INcRk

INcy

INcEd

xMxxM

kxNxxN

γχγχ

0.1905.0

0.1)()(

)()()()(

1,,,

,,

1,,

,

≤

≤+M

IMcRky

IMcLT

IMcEdy

zyM

INcRk

INcz

INcEd

xMxxM

kxNxxN

γχγχ

955.0

942.0

1

53.942)5()(

1.3022)0()(

752.0)5()(

1)0()(

634.0)0()(

300)(

1100)(

1

,,,

,

,

,

,

,,

,

=

=

=

==

==

==

==

==

=

=

zy

yy

M

elyIMcRky

RkINcRk

yIMcLT

yINcy

zINcz

IMcEdy

INcEd

kk

kNmMxM

kNNxN

x

x

x

kNmxM

kNxN

γ

χχ

χχ

χχ


NEd,max=1337.75 kN My,Ed,max= 364.84 kNm LF = 1.216

LF ≈ 1/0.905=1.105

diff%1.9→

Beam-columns

Research challenges





q  Generalization of the over-strength factors to any shape of cross-section / loading q  Utilization of the compressed flange

q  Calibrate equivalent imperfections e0/L for new imperfection factors and also for other types of cross section:

q  Properly account for local buckling due to shear / bending q  Development of direct reduction factor approach for beam-columns q  Other boundary conditions q  Partial restraints q  …

Research challenges





REFERENCES

Simões da Silva, L, Rimões, R and Gervásio, H - “Design of Steel Structures”, Ernst & Sohn and ECCS Press, 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, (2014, in print).

•  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=2 SemiComp Member Design – Design resistance of prismatic beam-columns”, Greiner et al, RFCS, 2011. http://www.steelconstruct.com

ECCS Steel Member Calculator – Design of columns, beams and beam-columns to EC3-1-1”, Silva et al, ECCS, 2011. http://www.steelconstruct.com

www.steelconstruct.com www.cmm.pt





ACKNOWLEDGEMENTS

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

[email protected] Gab. SF 5.11

http://steel.fsv.cvut.cz/suscos

ADVANCED DESIGN OF STEEL AND COMPOSITE STRUCTURES

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