Mathcad - EC3-LTB_10-01-20.xmcd

Column Item COx
Lateral torsional buckling
F06xx
Annex COxA
Page 1/7
Lateral Torsional Buckling (sections class 1-3) - Column Item COx
according to EN 1993-1-1:2005 (EC3)
section and eqn. numbers refer to this code
(Form EC3-LTB_06-04-18.mcd - adopted)
Profile chosen Profile := "HEA240"
Yield limit f y.k := 355 N
mm 2
Partial safety factor for material γ M := 1.1
f y.k
Design strength f y.d := f y.d = 323 N
γ M
mm 2
Young's modulus E 2.1⋅10 5 N
:=
mm 2
Section Properties
h profile
b profile
:= 230⋅mm
I y := 7763⋅cm 4 W y := 675⋅cm 3
:= 240⋅mm
I z := 2769⋅cm 4 W z := 231⋅cm 3
A profile := 76.8⋅cm 2 I T := 42⋅cm 4 I ω := 328500⋅cm 6
Select CLASS of cross-section CLASS := 2
Factor from elastic to plastic section modulus
if CLASS = 3 only elastic properties may be used
α pl.y := wenn( CLASS = 3 , 1,
1.14)
α pl.y = 1.14
α pl.z := wenn( CLASS = 3 , 1,
1.50)
α pl.z = 1.50
Design resistances of cross section
N pl.d
:= A profile ⋅f y.d
N pl.d = 2479kN
M pl.y.d
:= α pl.y ⋅W y ⋅f y.d
M pl.y.d = 248kNm
HS Augsburg - civil engineering
Baumgartner Str. 16, D-86161 Augsburg
+49(0) 821 - 55 86 - 3171, Fax - 3913
peter.knoedel@hs-augsburg.de
Released by: P. Knoedel
Date of release: 20.01.10
Printed: 20.01.2010 - 08:56
EC3-LTB_10-01-20.xmcd

Column Item COx
Lateral torsional buckling
F06xx
Annex COxA
Page 2/7
M pl.z.d
:= α pl.z ⋅W z ⋅f y.d
M pl.z.d = 112kNm
System length - distance of gable ends
L := 6.00m
L = 6.00 m
Actions
from excentrically connected beam:
(the moment from offset will be carried to the upper and lower
parts of the member to equal parts)
V y.Ed
:=
200kN
Moments about major axis
End moment M 1 (+/-):
M 1.y
:=
0.5⋅V y.Ed ⋅
⎛
⎜
⎝
h profile
2
+ 20mm
⎞
⎠
M 1.y
=
14kNm
End moment M 2 < M 1 (+/-): M 2.y := 0kNm
Relation of end moments
M 2.y
ψ y := ψ y = 0.000
M 1.y
Moment from transverse loads (positive only)
bild1
M 3.y
:=
150kNm
Position of Maximum (dist. from M2) a y := 1.5m
a y = 1500mm
⋅( )
a y
M y.s := M 2.y +
L M 1.y − M 2.y + M 3.y
M y.s = 153.4kNm
Design moment, maximum by absolute value
HS Augsburg - civil engineering
Baumgartner Str. 16, D-86161 Augsburg
+49(0) 821 - 55 86 - 3171, Fax - 3913
peter.knoedel@hs-augsburg.de
Released by: P. Knoedel
Date of release: 20.01.10
Printed: 20.01.2010 - 08:56
EC3-LTB_10-01-20.xmcd

Column Item COx
Lateral torsional buckling
F06xx
Annex COxA
Page 3/7
( )
M y.Ed := max M 1.y , M 2.y , M y.s M y.Ed = 153kNm
see Table B.3
⎛
⎜
⎝
M y.s
α y.s := wenn M y.s < M 1.y , , 0 α y.s = 0.000
⎛
⎜
⎝
M 1.y
M 1.y
α y.h := wenn M y.s > M 1.y , , 0 α y.h = 0.088
M y.s
⎞
⎠
⎞
⎠
Flexural buckling about minor axis
Factor for buckling length β z := 1.0
buckling length L cr.z := β z ⋅L
L cr.z = 6.00 m
π 2 EI ⋅ z
Bifurcation load N cr.z.d := ⋅
N cr.z.d = 1449kN
2
L γ cr.z M
N pl.d
normalised Slenderness λ bar.z := λ bar.z = 1.308
N cr.z.d
η p
h profile
:= η p = 0.96
b profile
Buckling curve (table 6.2) - ONLY rolled sections and t,f =< 40 mm
BC z := "b" if η p > 1.2
BC z = "c"
"c" if η p ≤ 1.2
Imperfection factor
(table 6.1)
α z := 0.34 if BC z = "b"
α z = 0.49
0.49 if BC z = "c"
Reduction factor according to eq. (6.49)
⎡
⎣
⋅( )
2
Φ z := 0.5⋅
1 + α z λ bar.z − 0.2 + λ bar.z Φ z = 1.627
⎤
⎦
HS Augsburg - civil engineering
Baumgartner Str. 16, D-86161 Augsburg
+49(0) 821 - 55 86 - 3171, Fax - 3913
peter.knoedel@hs-augsburg.de
Released by: P. Knoedel
Date of release: 20.01.10
Printed: 20.01.2010 - 08:56
EC3-LTB_10-01-20.xmcd

Column Item COx
Lateral torsional buckling
F06xx
Annex COxA
Page 4/7
χ z := 1 if λ bar.z ≤ 0.2
χ z = 0.386
1
2 2
Φ z + Φ z − λ bar.z
if
λ bar.z > 0.2
Design buckling resistance under normal compression
N b.z.Rd
:= χ z ⋅N pl.d
N b.z.Rd = 956kN
Elastic critical moment for lateral-torsional buckling
according to DIN 18800-2 Element 311, eq. 19:
2
I ω + 0.039⋅L cr.z ⋅I T
c := c = 18.2cm
Distance of centroid from attack of transverse forces;
positive, if a torsion about the centroid produces a restoring force
I z
input "t" for top, "m" for middle, "b" for bottom z := "t"
z p
h profile
:= − if z = "t"
z p = −115mm
2
0mm if z = "m"
h profile
2
if
z =
"b"
HS Augsburg - civil engineering
Baumgartner Str. 16, D-86161 Augsburg
+49(0) 821 - 55 86 - 3171, Fax - 3913
peter.knoedel@hs-augsburg.de
Released by: P. Knoedel
Date of release: 20.01.10
Printed: 20.01.2010 - 08:56
EC3-LTB_10-01-20.xmcd

Column Item COx
Lateral torsional buckling
F06xx
Annex COxA
Page 5/7
Moment coefficient according to
DIN 18800-2 Table 10
ζ := 1.77
bild3
Critical Moment for LTB (DIN 18800-2 Element 311, eq. 19):
M cr.y.d
⎛
⎝
ζ⋅N cr.z.d c 2 2
:= ⋅ + 0.25⋅z p + 0.5⋅z p M cr.y.d = 342kNm
⎞
⎠
normalised slenderness (eq. (6.56))
λ bar.LT
M pl.y.d
:= λ bar.LT = 0.852
M cr.y.d
Buckling curve (table 6.5) - ONLY rolled sections
BCLT := "b" if η p ≤ 2
BCLT = "b"
"c" if η p > 2
Imperfection factor
(table 6.3)
Reduction factor according to eq. (6.57)
α LT := 0.34 if BCLT = "b"
α LT = 0.34
0.49 if BCLT = "c"
Parameters λ bar.LT.0 := 0.4
β := 0.75
According to 6.3.2.2 (4) LTB may be ignored, if λ,LT / λ,LT,0

Column Item COx
Lateral torsional buckling
F06xx
Annex COxA
Page 6/7
⎡
⎣
⋅( )
M y.Ed
M cr.y.d ⋅γ M
1
⋅ = 2.545
2
λ bar.LT.0
2
Φ LT := 0.5⋅
1 + α LT λ bar.LT − λ bar.LT.0 + β⋅λ bar.LT Φ LT = 0.849
⎤
⎦
1
χ LT := χ LT = 0.788
2
2
Φ LT + Φ LT − βλ ⋅ bar.LT
⎛
⎜
⎝
1
checking limits χ LT := min χ LT , 1.0,
χ LT = 0.788
2
λ bar.LT
Correction factor from table 6.6
⎞
⎠
k c := 0.82
Modification factor eq. (6.58)
( )
⎡
⎣
⋅( ) 2
f := 1 − 0.5⋅ 1 − k c ⋅ 1 − 2.0 λ bar.LT − 0.8
f := min( f,
1)
f = 0.91
⎤
⎦
bild2
Design buckling resistance in bending
M b.Rd
Modified reduction factor
χ LT.mod
⎛
min χ LT
:= ⎜ , 1 χ LT.mod = 0.866
f
⎝
:= χ LT.mod ⋅M pl.y.d
M b.Rd = 215kNm
⎞
⎠
k zy := 1
k yy := 1
Stability check according to eq. (6.61)
utilisation of flexural buckling about major axis by N
HS Augsburg - civil engineering
Baumgartner Str. 16, D-86161 Augsburg
+49(0) 821 - 55 86 - 3171, Fax - 3913
peter.knoedel@hs-augsburg.de
Released by: P. Knoedel
Date of release: 20.01.10
Printed: 20.01.2010 - 08:56
EC3-LTB_10-01-20.xmcd

Column Item COx
Lateral torsional buckling
F06xx
Annex COxA
Page 7/7
M y.Ed
utilisation of LTB by My η 1.My := ⋅k yy
η 1.My = 0.713
M b.Rd
Stability check according to eq. (6.62)
M y.Ed
utilisation of LTB by My η 2.My := ⋅k zy
η 2.My = 0.713
M b.Rd
Utilisation in cross section check (without interaction):
utilisation by My:
η My
M y.Ed
:= η My = 0.618
M pl.y.d
HS Augsburg - civil engineering
Baumgartner Str. 16, D-86161 Augsburg
+49(0) 821 - 55 86 - 3171, Fax - 3913
peter.knoedel@hs-augsburg.de
Released by: P. Knoedel
Date of release: 20.01.10
Printed: 20.01.2010 - 08:56
EC3-LTB_10-01-20.xmcd