ASEN3112.Lect09.Slides

7/30/2019 ASEN3112.Lect09.Slides

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ASEN 3112 - Structures

Torsion Of

Closed Thin Wall(CTW) Sections

9

ASEN 3112 Lecture 9 – Slide 1



Recall the Classification of Thin Wall (TW)

Cross Sections Under Torque


Closed Thin Wall (CTW) Sections

at least one cell shear flow circuit can be established

Single Cell: just one cell

Multicell: more than one cell

Open Thin Wall (OTW) Sectionsno cell shear flow circuit can be established

Hybrid Thin Wall (HTW) Sections

contains both OTW and CTW components




Sample CTW Cross Sections


tube boxaircraftfuselage

space shuttlefuselage

car frame (unibodyconstruction) wing torque box




Single-Cell CTW Section


(b)(a)

z

T

yt = t(s)

s

(c)

Enclosed

area= T

Shear flow

2A E A E

h

O

q ds

ds




Shear Stress Formula


T = dT = q. h ds = q h ds = q 2 dA E = q. 2 A E = 2 q A E

q =T

2 A E

τ =q

t =

T

2t A E

τmax =q

t mi n

=T

2t mi n A E

Equilibrium of internal torque T with the assumed shear stress distribution

(constant across wall thickness, directed along tangent to midline) gives

in which A is the enclosed area defined by the wall midline (which

should be confused with the cross section area or material area)

Solving for q gives the shear flow and wall stress formula

Since q is constant along the contour, plainly the maximum shear stress

occurs where the wall thickness t is minimum:

E




Twist Angle Formula


It can be shown that the twist-angle rate is given by

Here G is the shear modulus. Derivation of this formula requires the

use of energy methods in mechanics, which is a graduate-level topic.

If the wall thickness t is constant along the cell contour, the foregoing

formula simplifies to

=

dx=

q p

2 A E t =

T

G J in which

in which

J =4 A2

E t

p

φ'

φ'

d φ

d φ

=

dx=

q

2G A E

ds

t =

T

4G A2 E

ds

t =

T

G J J =

4 A2 E

dst

where p = ds is the midline perimeter length, or simply the perimeter.




Multicell CTW Sections

Are Not Treated in This Course


(b)(a)t

1

2

3

q1

q2

q3

C C B D D

(c)q

1

q3

q1

q2

_

q3q2_

q2

q2T

A A

B

z

y

Cell flows Wall flows




Worked Out Example Cross Sections


T

d/2

d

d

t

t

t

t

t

T T

Same d and t as in (a)

thin cut

t

(a) (c)(b)

d

d/2

d

CTW section:rectangular tube

Hybrid section:CTW + OTW (2)

OTW section:slitted rectangular tube

All d 's are midline dimensions




Torqued Hybrid TW (HTW) "A" Section

For Recitation


b1

t /21

t 1 t 1

b1

b2b3

60 60

60b1

t 2 t 3

A

B C

D E

A

B

B

C

C

D E

1

2 3

decompositionCTW

OTW OTW

(b)(a)

T

b1 = 200 mm, b2 = b3 = 400 mm, t 1 = t 2 = t 3 = 6 mm

G = 80 GPa (steel), T = 604 103 mm-N, L = 2514 mm


ASEN3112.Lect09.Slides

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