Mathcad - EC3-LTB_10-01-20.xmcd
Mathcad - EC3-LTB_10-01-20.xmcd 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
- Page 2 and 3: Column Item COx Lateral torsional b
- Page 4 and 5: Column Item COx Lateral torsional b
- Page 6 and 7: Column Item COx Lateral torsional b
Column Item COx<br />
Lateral torsional buckling<br />
F06xx<br />
Annex COxA<br />
Page 1/7<br />
Lateral Torsional Buckling (sections class 1-3) - Column Item COx<br />
according to EN 1993-1-1:2005 (<strong>EC3</strong>)<br />
section and eqn. numbers refer to this code<br />
(Form <strong>EC3</strong>-<strong>LTB</strong>_06-04-18.mcd - adopted)<br />
Profile chosen Profile := "HEA240"<br />
Yield limit f y.k := 355 N<br />
mm 2<br />
Partial safety factor for material γ M := 1.1<br />
f y.k<br />
Design strength f y.d := f y.d = 323 N<br />
γ M<br />
mm 2<br />
Young's modulus E 2.1⋅<strong>10</strong> 5 N<br />
:=<br />
mm 2<br />
Section Properties<br />
h profile<br />
b profile<br />
:= 230⋅mm<br />
I y := 7763⋅cm 4 W y := 675⋅cm 3<br />
:= 240⋅mm<br />
I z := 2769⋅cm 4 W z := 231⋅cm 3<br />
A profile := 76.8⋅cm 2 I T := 42⋅cm 4 I ω := 328500⋅cm 6<br />
Select CLASS of cross-section CLASS := 2<br />
Factor from elastic to plastic section modulus<br />
if CLASS = 3 only elastic properties may be used<br />
α pl.y := wenn( CLASS = 3 , 1,<br />
1.14)<br />
α pl.y = 1.14<br />
α pl.z := wenn( CLASS = 3 , 1,<br />
1.50)<br />
α pl.z = 1.50<br />
Design resistances of cross section<br />
N pl.d<br />
:= A profile ⋅f y.d<br />
N pl.d = 2479kN<br />
M pl.y.d<br />
:= α pl.y ⋅W y ⋅f y.d<br />
M pl.y.d = 248kNm<br />
HS Augsburg - civil engineering<br />
Baumgartner Str. 16, D-86161 Augsburg<br />
+49(0) 821 - 55 86 - 3171, Fax - 3913<br />
peter.knoedel@hs-augsburg.de<br />
Released by: P. Knoedel<br />
Date of release: 20.<strong>01</strong>.<strong>10</strong><br />
Printed: 20.<strong>01</strong>.20<strong>10</strong> - 08:56<br />
<strong>EC3</strong>-<strong>LTB</strong>_<strong>10</strong>-<strong>01</strong>-<strong>20.xmcd</strong>
Column Item COx<br />
Lateral torsional buckling<br />
F06xx<br />
Annex COxA<br />
Page 2/7<br />
M pl.z.d<br />
:= α pl.z ⋅W z ⋅f y.d<br />
M pl.z.d = 112kNm<br />
System length - distance of gable ends<br />
L := 6.00m<br />
L = 6.00 m<br />
Actions<br />
from excentrically connected beam:<br />
(the moment from offset will be carried to the upper and lower<br />
parts of the member to equal parts)<br />
V y.Ed<br />
:=<br />
200kN<br />
Moments about major axis<br />
End moment M 1 (+/-):<br />
M 1.y<br />
:=<br />
0.5⋅V y.Ed ⋅<br />
⎛<br />
⎜<br />
⎝<br />
h profile<br />
2<br />
+ 20mm<br />
⎞<br />
⎠<br />
M 1.y<br />
=<br />
14kNm<br />
End moment M 2 < M 1 (+/-): M 2.y := 0kNm<br />
Relation of end moments<br />
M 2.y<br />
ψ y := ψ y = 0.000<br />
M 1.y<br />
Moment from transverse loads (positive only)<br />
bild1<br />
M 3.y<br />
:=<br />
150kNm<br />
Position of Maximum (dist. from M2) a y := 1.5m<br />
a y = 1500mm<br />
⋅( )<br />
a y<br />
M y.s := M 2.y +<br />
L M 1.y − M 2.y + M 3.y<br />
M y.s = 153.4kNm<br />
Design moment, maximum by absolute value<br />
HS Augsburg - civil engineering<br />
Baumgartner Str. 16, D-86161 Augsburg<br />
+49(0) 821 - 55 86 - 3171, Fax - 3913<br />
peter.knoedel@hs-augsburg.de<br />
Released by: P. Knoedel<br />
Date of release: 20.<strong>01</strong>.<strong>10</strong><br />
Printed: 20.<strong>01</strong>.20<strong>10</strong> - 08:56<br />
<strong>EC3</strong>-<strong>LTB</strong>_<strong>10</strong>-<strong>01</strong>-<strong>20.xmcd</strong>
Column Item COx<br />
Lateral torsional buckling<br />
F06xx<br />
Annex COxA<br />
Page 3/7<br />
( )<br />
M y.Ed := max M 1.y , M 2.y , M y.s M y.Ed = 153kNm<br />
see Table B.3<br />
⎛<br />
⎜<br />
⎝<br />
M y.s<br />
α y.s := wenn M y.s < M 1.y , , 0 α y.s = 0.000<br />
⎛<br />
⎜<br />
⎝<br />
M 1.y<br />
M 1.y<br />
α y.h := wenn M y.s > M 1.y , , 0 α y.h = 0.088<br />
M y.s<br />
⎞<br />
⎠<br />
⎞<br />
⎠<br />
Flexural buckling about minor axis<br />
Factor for buckling length β z := 1.0<br />
buckling length L cr.z := β z ⋅L<br />
L cr.z = 6.00 m<br />
π 2 EI ⋅ z<br />
Bifurcation load N cr.z.d := ⋅<br />
N cr.z.d = 1449kN<br />
2<br />
L γ cr.z M<br />
N pl.d<br />
normalised Slenderness λ bar.z := λ bar.z = 1.308<br />
N cr.z.d<br />
η p<br />
h profile<br />
:= η p = 0.96<br />
b profile<br />
Buckling curve (table 6.2) - ONLY rolled sections and t,f =< 40 mm<br />
BC z := "b" if η p > 1.2<br />
BC z = "c"<br />
"c" if η p ≤ 1.2<br />
Imperfection factor<br />
(table 6.1)<br />
α z := 0.34 if BC z = "b"<br />
α z = 0.49<br />
0.49 if BC z = "c"<br />
Reduction factor according to eq. (6.49)<br />
⎡<br />
⎣<br />
⋅( )<br />
2<br />
Φ z := 0.5⋅<br />
1 + α z λ bar.z − 0.2 + λ bar.z Φ z = 1.627<br />
⎤<br />
⎦<br />
HS Augsburg - civil engineering<br />
Baumgartner Str. 16, D-86161 Augsburg<br />
+49(0) 821 - 55 86 - 3171, Fax - 3913<br />
peter.knoedel@hs-augsburg.de<br />
Released by: P. Knoedel<br />
Date of release: 20.<strong>01</strong>.<strong>10</strong><br />
Printed: 20.<strong>01</strong>.20<strong>10</strong> - 08:56<br />
<strong>EC3</strong>-<strong>LTB</strong>_<strong>10</strong>-<strong>01</strong>-<strong>20.xmcd</strong>
Column Item COx<br />
Lateral torsional buckling<br />
F06xx<br />
Annex COxA<br />
Page 4/7<br />
χ z := 1 if λ bar.z ≤ 0.2<br />
χ z = 0.386<br />
1<br />
2 2<br />
Φ z + Φ z − λ bar.z<br />
if<br />
λ bar.z > 0.2<br />
Design buckling resistance under normal compression<br />
N b.z.Rd<br />
:= χ z ⋅N pl.d<br />
N b.z.Rd = 956kN<br />
Elastic critical moment for lateral-torsional buckling<br />
according to DIN 18800-2 Element 311, eq. 19:<br />
2<br />
I ω + 0.039⋅L cr.z ⋅I T<br />
c := c = 18.2cm<br />
Distance of centroid from attack of transverse forces;<br />
positive, if a torsion about the centroid produces a restoring force<br />
I z<br />
input "t" for top, "m" for middle, "b" for bottom z := "t"<br />
z p<br />
h profile<br />
:= − if z = "t"<br />
z p = −115mm<br />
2<br />
0mm if z = "m"<br />
h profile<br />
2<br />
if<br />
z =<br />
"b"<br />
HS Augsburg - civil engineering<br />
Baumgartner Str. 16, D-86161 Augsburg<br />
+49(0) 821 - 55 86 - 3171, Fax - 3913<br />
peter.knoedel@hs-augsburg.de<br />
Released by: P. Knoedel<br />
Date of release: 20.<strong>01</strong>.<strong>10</strong><br />
Printed: 20.<strong>01</strong>.20<strong>10</strong> - 08:56<br />
<strong>EC3</strong>-<strong>LTB</strong>_<strong>10</strong>-<strong>01</strong>-<strong>20.xmcd</strong>
Column Item COx<br />
Lateral torsional buckling<br />
F06xx<br />
Annex COxA<br />
Page 5/7<br />
Moment coefficient according to<br />
DIN 18800-2 Table <strong>10</strong><br />
ζ := 1.77<br />
bild3<br />
Critical Moment for <strong>LTB</strong> (DIN 18800-2 Element 311, eq. 19):<br />
M cr.y.d<br />
⎛<br />
⎝<br />
ζ⋅N cr.z.d c 2 2<br />
:= ⋅ + 0.25⋅z p + 0.5⋅z p M cr.y.d = 342kNm<br />
⎞<br />
⎠<br />
normalised slenderness (eq. (6.56))<br />
λ bar.LT<br />
M pl.y.d<br />
:= λ bar.LT = 0.852<br />
M cr.y.d<br />
Buckling curve (table 6.5) - ONLY rolled sections<br />
BCLT := "b" if η p ≤ 2<br />
BCLT = "b"<br />
"c" if η p > 2<br />
Imperfection factor<br />
(table 6.3)<br />
Reduction factor according to eq. (6.57)<br />
α LT := 0.34 if BCLT = "b"<br />
α LT = 0.34<br />
0.49 if BCLT = "c"<br />
Parameters λ bar.LT.0 := 0.4<br />
β := 0.75<br />
According to 6.3.2.2 (4) <strong>LTB</strong> may be ignored, if λ,LT / λ,LT,0
Column Item COx<br />
Lateral torsional buckling<br />
F06xx<br />
Annex COxA<br />
Page 6/7<br />
⎡<br />
⎣<br />
⋅( )<br />
M y.Ed<br />
M cr.y.d ⋅γ M<br />
1<br />
⋅ = 2.545<br />
2<br />
λ bar.LT.0<br />
2<br />
Φ LT := 0.5⋅<br />
1 + α LT λ bar.LT − λ bar.LT.0 + β⋅λ bar.LT Φ LT = 0.849<br />
⎤<br />
⎦<br />
1<br />
χ LT := χ LT = 0.788<br />
2<br />
2<br />
Φ LT + Φ LT − βλ ⋅ bar.LT<br />
⎛<br />
⎜<br />
⎝<br />
1<br />
checking limits χ LT := min χ LT , 1.0,<br />
χ LT = 0.788<br />
2<br />
λ bar.LT<br />
Correction factor from table 6.6<br />
⎞<br />
⎠<br />
k c := 0.82<br />
Modification factor eq. (6.58)<br />
( )<br />
⎡<br />
⎣<br />
⋅( ) 2<br />
f := 1 − 0.5⋅ 1 − k c ⋅ 1 − 2.0 λ bar.LT − 0.8<br />
f := min( f,<br />
1)<br />
f = 0.91<br />
⎤<br />
⎦<br />
bild2<br />
Design buckling resistance in bending<br />
M b.Rd<br />
Modified reduction factor<br />
χ LT.mod<br />
⎛<br />
min χ LT<br />
:= ⎜ , 1 χ LT.mod = 0.866<br />
f<br />
⎝<br />
:= χ LT.mod ⋅M pl.y.d<br />
M b.Rd = 215kNm<br />
⎞<br />
⎠<br />
k zy := 1<br />
k yy := 1<br />
Stability check according to eq. (6.61)<br />
utilisation of flexural buckling about major axis by N<br />
HS Augsburg - civil engineering<br />
Baumgartner Str. 16, D-86161 Augsburg<br />
+49(0) 821 - 55 86 - 3171, Fax - 3913<br />
peter.knoedel@hs-augsburg.de<br />
Released by: P. Knoedel<br />
Date of release: 20.<strong>01</strong>.<strong>10</strong><br />
Printed: 20.<strong>01</strong>.20<strong>10</strong> - 08:56<br />
<strong>EC3</strong>-<strong>LTB</strong>_<strong>10</strong>-<strong>01</strong>-<strong>20.xmcd</strong>
Column Item COx<br />
Lateral torsional buckling<br />
F06xx<br />
Annex COxA<br />
Page 7/7<br />
M y.Ed<br />
utilisation of <strong>LTB</strong> by My η 1.My := ⋅k yy<br />
η 1.My = 0.713<br />
M b.Rd<br />
Stability check according to eq. (6.62)<br />
M y.Ed<br />
utilisation of <strong>LTB</strong> by My η 2.My := ⋅k zy<br />
η 2.My = 0.713<br />
M b.Rd<br />
Utilisation in cross section check (without interaction):<br />
utilisation by My:<br />
η My<br />
M y.Ed<br />
:= η My = 0.618<br />
M pl.y.d<br />
HS Augsburg - civil engineering<br />
Baumgartner Str. 16, D-86161 Augsburg<br />
+49(0) 821 - 55 86 - 3171, Fax - 3913<br />
peter.knoedel@hs-augsburg.de<br />
Released by: P. Knoedel<br />
Date of release: 20.<strong>01</strong>.<strong>10</strong><br />
Printed: 20.<strong>01</strong>.20<strong>10</strong> - 08:56<br />
<strong>EC3</strong>-<strong>LTB</strong>_<strong>10</strong>-<strong>01</strong>-<strong>20.xmcd</strong>