4. Torsion - FSv ČVUT v Praze -- FSv ČVUT --...
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©4 Prof. Ing. Josef Macháček, DrSc. OK3 1 4. Torsion Open and closed cross sections, simple St. Venant and warping torsion, interaction of bending and torsion, Eurocode approach. Common is elastic solution (nonlinear plastic analysis – e.g. Strelbickaja) Eurocode 3 enables combination of plastic bending moment and elastic torsion. Should be distinguished: - simple torsion: only shear stresses arise, - warping torsion: both shear and direct (normal) stresses arise. 1. Open cross sections (e.g. I, U, L) a) Simple (Saint Venant’s) torsion (occurs only exceptionally, see later) τ t τ i T t t b i = 1 i = 2 i = 3 Only shear arises: 3 / 0 M y (i) t t t(i) γ τ f t I T ≤ = (maximal in t max ) ∑ = i 3 i i t 3 1 t b I α (influence of rounding of rolled sections, otherwise = 1)
Transcript of 4. Torsion - FSv ČVUT v Praze -- FSv ČVUT --...
© 4 Prof. Ing. Josef Macháček, DrSc.
OK3 1
4. TorsionOpen and closed cross sections, simple St. Venant and warping torsion,interaction of bending and torsion, Eurocode approach.
Common is elastic solution (nonlinear plastic analysis – e.g. Strelbickaja)Eurocode 3 enables combination of plastic bending moment and elastic torsion.
Should be distinguished:- simple torsion: only shear stresses arise,- warping torsion: both shear and direct (normal) stresses arise.
1. Open cross sections (e.g. I, U, L)a) Simple (Saint Venant’s) torsion (occurs only exceptionally, see later)
τt
τiTt
t
bi = 1
i = 2
i = 3
Only shear arises:
3/ 0My
(i)t
tt(i)
γτ
ft
IT
≤=
(maximal in tmax)
∑=i
3iit 3
1 tbI α
(influence of rounding of rolled sections,otherwise = 1)
© 4 Prof. Ing. Josef Macháček, DrSc.
OK3 2
b) Warping torsion (according to Vlasov’s theory)
• one part of a torsion moment T is transmitted by simple torsion Tt , • other part by bending torsion Tw : T = Tt + Tw
Assumptions: 1. Rigid cross section,2. Null shear deformation
(shear lag ignored).
moment of simple torsion
moment of bending torsion
bimoment
bending torsion
T
S shear centre(bending centre)
σwτt τw
= +++
+
-
-
in warping torsion everythingis related to central line
internal forces: Tt Tw B
© 4 Prof. Ing. Josef Macháček, DrSc.
OK3 3
Shear stresses:• simple torsion τt
• bending torsion
e.g.:
Direct (normal) stress:
Resulting stresses:
www W
BIwB
==σ
w
www It
ST=τ
first sectorial moment
warping constant
t
τw,max τt
t
stress through thickness t
sectorial section modulus
Applies „bending analogy“ : B → M or σw → σ
Tw→ V or τw → τ
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OK3 4
Sectorial characteristics
• Rolled sections – see tables.
• In general from sectorial coordinate:
I cross section:
4ds
s
hbrw =∫=
z
2
A
2w 4
dA IhwI == ∫U cross section:
a = a
w Swin this positionno torsion !!
GS
The main sectorial coordinate:
First sectorial moment:
16dA
2
Aw
tbhwS =∫=
Second sectorial moment (warping constant):
w, Sw, Iw ... see tables
0dAdAAA
=∫=∫ wzwyPosition of S:
(product sectorial moments)
t
h
b
wSw
rG ≡ S
z
© 4 Prof. Ing. Josef Macháček, DrSc.
OK3 5
Determination of internal forces due to torsion:• solution of Vlasov’s differential equations, or directly from formulas,• based on „bending analogy“.
Distribution of torsion moment:
eF
V
simple support in torsion (couple of forces)
M
Abending
torsion
T = Ve
B
Distribution of torsional moment due to eccentrical force corresponds to distribution of transverse force ateccentricity.Part transmitted by simple torsion is set aside:
κeVT ≅t
( )κ−≅ 1w eVT
( )κ−≅ 1eMBMe
Superposition for more complex loading is necessary:
κ ... see table of Eurocode Czech NA
e2
e1
- e3
© 4 Prof. Ing. Josef Macháček, DrSc.
OK3 6
Simplified (conservative) solution – neglects simple torsion:
bending of flanges only = bimoment(often adequate: it is conservative from σw point of view)
Important notes:
1. Large direct stresses, they can not be ignored!!
2. Direct stresses (warping torsion) do not arise:a) for loading by stresses τt, roughly also due to end
T loading (simple torsion arises only):
b) in sections composed of radiating outstands(because of w = Iw = 0):
3. In practice usually occurs „torsion about enforced axis“ (V):
T0
T T/h
T/h
h
(shear centre S is in cross point)
moreover, torsion is usually restricted by cladding rigidity ⇒ torsion may often be ignored.
analysis about original shear centre Sis uneconomical !!! S
S ≡ V e
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OK3 7
a) Simple torsion (shear stresses only, usual design)
2. Closed cross sections (e.g. )
Bredt’s shear flow (τt t ) = const.
(i)t(i) t
Tt
Ωτ =
b) Warping torsion: - Umanskij’s theory (rigid cross section),- Vlasov’s theory with non rigid cross section,- FEM (including influence of bevelled cross section,
gives also transversal bending moments in plates).
The stresses are the same as in open cross sections: τt, τw, σw. However, τw, σw are very small, commonly ignored even for bridges.
= 2 As
As
τt tidi
Tt
Contrary to open cross section the maximal shear τt
is in the thinnest plate and along thickness constant !!
© 4 Prof. Ing. Josef Macháček, DrSc.
OK3 8
3. Interaction of bending and torsion ( My + T )e
Direct stresses (open cross sections only):
M1ywMy γσσσ /f≤+=
M1yw
Ed
yLT
Edy, γχ
/fWB
WM
≤+
Shear stresses:
i.e.
( ) ( ) Rdpl,My
Edw,
M0y
Edt,RdT,pl, V
//f//f,V
⎥⎥⎦
⎤
⎢⎢⎣
⎡−−=
0332511
γτ
γτ
1≤RdT,pl,
Ed
VV Vpl,T,Rd is design plastic shear resistance of the cross section.
For open sections I and U
For closed sections ( ) Rdpl,My
Edt,RdT,pl, V
//fV
⎥⎥⎦
⎤
⎢⎢⎣
⎡−=
031
γτ
in U sections only
In general, bending and torsion stresses may be summed and von Mises criterionapplied:
132
000
22
0≤⎟
⎟⎠
⎞⎜⎜⎝
⎛+⎟
⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛−⎟
⎟⎠
⎞⎜⎜⎝
⎛+⎟
⎟⎠
⎞⎜⎜⎝
⎛
My
Ed
My
Edz,
My
Edx,
M0y
Edz,
My
Edx,
γτ
γσ
γσ
γσ
γσ
/f/f/f/f/f
