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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
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
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
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
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
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
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
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
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
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
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
Non-uniform members – approaches and problems
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
q Non-uniform members – Clauses 6.3.1 to 6.3.3 apply ???
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
Non-uniform members – approaches and problems
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
q Non-uniform members – Clauses 6.3.1 to 6.3.3 apply ???
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!
Non-uniform members – approaches and problems
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
q Non-uniform members – Clauses 6.3.1 to 6.3.3 apply ???
q Variation of cross section class
q Definition of an equivalent class for the member
Class 3Class 2 Class 1
My,Ed
NEd <<< Afy
Non-uniform members – approaches and problems
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
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
Non-uniform members – approaches and problems
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
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
Non-uniform members – approaches and problems
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
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
Non-uniform members – approaches and problems
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
q Non-uniform members – GENERAL METHOD (clause 6.3.4)
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
Non-uniform members – approaches and problems
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
q Non-uniform members – GENERAL METHOD (clause 6.3.4)
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
Non-uniform members – approaches and problems
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
q Non-uniform members – GENERAL METHOD (clause 6.3.4)
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
Non-uniform members – approaches and problems
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
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
Non-uniform members – approaches and problems
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
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
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
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
Tapered columns Differential equation
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
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
Tapered columns Differential equation
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
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
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
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
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
⎥⎦
⎤⎢⎣
⎡ ′′−⎥⎦
⎤⎢⎣
⎡
−+=
)(.))().((
)()(
)()(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
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
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
Design resistance of tapered columns and beams
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
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
Design resistance of tapered columns and beams
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
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
Design resistance of tapered columns and beams
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
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
Design resistance of tapered columns and beams
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
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
Design resistance of tapered columns and beams
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
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
Design resistance of tapered columns and beams
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
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
Design resistance of tapered columns and beams
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
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
Design resistance of tapered columns and beams
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
2. Application of the method
COLUMNS – DESIGN METHODOLOGY
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
Design resistance of tapered columns and beams
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
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 :
COLUMNS – DESIGN METHODOLOGY
( )( )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.
Design resistance of tapered columns and beams
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
q Necessary parameters
q Imperfection factor αy or αz
COLUMNS – DESIGN METHODOLOGY
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
Design resistance of tapered columns and beams
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
q Necessary parameters
q Overstrength factor φy or φz
COLUMNS – DESIGN METHODOLOGY
FB out-of-plane FB in-plane
⎥⎦
⎤⎢⎣
⎡ −++
h
hhw
Aht
γγγ
10)1)(41(
1min
111
min
min
+−+
h
hw
Ath
γγ
Design resistance of tapered columns and beams
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
q Results q Out-of-plane
(γh=hmax/hmin=1.8)
COLUMNS – DESIGN METHODOLOGY
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)
Design resistance of tapered columns and beams
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
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!
COLUMNS – DESIGN METHODOLOGY
Design resistance of tapered columns and beams
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
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
Design resistance of tapered columns and beams
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
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=
Design resistance of tapered columns and beams
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
q Minor axis verification
q Reduction factor
q Verification
q Numerical analysis, GMNIA
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
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,, →=
Design resistance of tapered columns and beams
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
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%
Design resistance of tapered columns and beams
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
q Major 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
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
Design resistance of tapered columns and beams
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
q Major axis verification
q Reduction factor
q Verification
q Numerical analysis, GMNIA
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
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,, →=
Design resistance of tapered columns and beams
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
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
(…)
Design resistance of tapered columns and beams
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
2. Application of the method
BEAMS – DESIGN METHODOLOGY
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
Design resistance of tapered columns and beams
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
q Necessary parameters
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
BEAMS – DESIGN METHODOLOGY
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≤
Design resistance of tapered columns and beams
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
q Necessary parameters
q xc,limII
BEAMS – DESIGN METHODOLOGY
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
Design resistance of tapered columns and beams
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
q Necessary parameters
q Overstrength factor φLT
BEAMS – DESIGN METHODOLOGY
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|
Design resistance of tapered columns and beams
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
q Necessary parameters
q Overstrength factor φLT
BEAMS – DESIGN METHODOLOGY
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
Design resistance of tapered columns and beams
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
q Results
q Possible problems: q Web buckling
BEAMS – DESIGN METHODOLOGY
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
Design resistance of tapered columns and beams
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
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
Design resistance of tapered columns and beams
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
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
Design resistance of tapered columns and beams
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
q Calculation of slenderness at x=xcI
q Second order failure location, x=xc,limII
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
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
Design resistance of tapered columns and beams
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
q Over-strength factor, φ
q Determination of imperfection, η
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
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
Design resistance of tapered columns and beams
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
q Reduction factor
q Verification
q Numerical analysis, GMNIA
kNmxMkNxMxM IcEdy
IcRdy
IcRdb 300)(45.82519.1097751.0)()( ,,, =>=×== χ
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
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, →=
Design resistance of tapered columns and beams
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
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
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
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 L3 NO NO NO NO Member stability procedures Global Lcr,z Lcr,y
Buckling lengthCheckLocal imperfections
+2nd order effects P-‐δ
Material nonlinearity
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
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
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 L3 NO NO NO NO Member stability procedures Global Lcr,z Lcr,y
Buckling lengthCheckLocal imperfections
+2nd order effects P-‐δ
Material nonlinearity
q Difficulties for each approach
Calibrate e0/L for new imperfection factors Need to develop
àIn-plane àOut-of-plane
Approaches: àInteraction àGeneralized
slenderness
Beam-columns
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
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 L3 NO NO NO NO Member stability procedures Global Lcr,z Lcr,y
Buckling lengthCheckLocal imperfections
+2nd order effects P-‐δ
Material nonlinearity
q Difficulties for each approach
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
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
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
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
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
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
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
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
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
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
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
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
q Interaction formulae
q Data:
BEAM-COLUMNS - EXAMPLE
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
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
q Cmy, CmLT
BEAM-COLUMNS - EXAMPLE
( ) ( ) 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
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
q Verification
BEAM-COLUMNS - EXAMPLE
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
q Numerical analysis, GMNIA
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
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
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
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
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
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
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
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
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
Module A – Design of non-uniform members
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards
and Catastrophic Events
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.
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