torsion vlasov muy buena

8/12/2019 Torsion Vlasov Muy Buena

1/8

4 Prof. Ing. Josef Machek, 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 Venants) torsion (occurs only exceptionally, see later)

t

iTt

t

b i= 1

i= 2

i= 3

Only shear arises:

3

/ 0My(i)

t

tt(i)

ftI

T (maximal in tmax)

i

3iit

31 tbI

(influence of rounding of rolled sections,

otherwise = 1)


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OK3 2

b) Warping tors ion (according to Vlasovs theory)

one part of a torsion moment Tis 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)

wt w

= ++

+

+

-

-

in warping torsion everything

is related to central line

internal forces: Tt Tw B


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OK3 3

Shear stresses:

simple torsion t

bending torsion

e.g.:

Direct (normal) stress:

Result ing stresses:

ww

wW

B

I

wB =

w

www

It

ST

=first sectorial moment

warping constant

t

w,max tt

stress through th ickness t

sectorial section modulus

Applies bending analogy : B M or w Tw V or w


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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

4dA I

hwI =U cross section:

a = a

w Swin this position

no 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

r

G S

z


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OK3 5

Determination of internal forces due to torsion:

solution of Vlasovs 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 dis tribution of transverse force at

eccentricity.Part transmitted by simple torsion is set aside:

eVTt 1w eVT1eMBMe

Superposition for more complex loading is necessary:

... see table of EurocodeCzech NA

e2

e1

- e3


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OK3 6

Simplified (conservative) solution neglects simple torsion:

bending of flanges only = bimoment

(often adequate: it is conservative

from w 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 t, roughly also due to endTloading (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):

T

0

T T/h

T/h

h

(shear centre Sis in cross point)

moreover, torsion is usually restricted by cladding rigidity torsion may often be ignored.

analysis about original shear centre S

is 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. )

Bredts shear flow (t t ) = const .

(i)t(i)

t

Tt

=

b) Warping torsion: - Umanskijs theory (rigid cross section),- Vlasovs 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: t, w, w.However, w, w are very small, commonly ignored even for br idges.

= 2As

As

t

tidi

Tt

Contrary to open cross section the maximal sheart

is in the thinnest plate and along thickness constant !!


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OK3 8

3. Interaction of bending and torsion ( My + T)e

Direct stresses (open cross sections only):

M1ywMy /fM1y

w

Ed

yLT

Edy, /fWB

W

M Shear stresses:

i.e.

Rdpl,My Edw,M0y Edt,RdT,pl, V//f//f,V

0332511

1RdT,pl,

Ed

V

VVpl,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

in U sections only

In general, bending and torsion stresses may be summed and von Mises criterion

applied:13

2

000

22

0

My

Ed

My

Edz,

My

Edx,

M0y

Edz,

My

Edx,

/f/f/f/f/f

torsion vlasov muy buena

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