Department of Continuum Mechanics and Structural...

Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero

Bachelor in Aerospace Engineering

Department of Continuum Mechanics and Structural Engineering

Aerospace Structures

Chapter 2. Bending, shear and torsion of thin-walled beams Bending and shear of open and closed, thin-walled beams (II)


OBJETIVES

Objetives

• Define and determine the shear centre

• Estimate the shear stress distribution by a alternate method to theproposed in section 2

• Calculate the shear stress distribution on multiple cell sections

CHAPTER 2. Bending, shear and torsion of thin-walled beams

Bending and shear of open and closed, thin-walled beams

(II)


Index

CHAPTER 2. Bending, shear and torsion of thin-walled beams

Bending and shear of open and closed, thin-walled beams

(II)

1. Concept of shear centre2. Alternate method to calculate shear stress3. Stress field on multiple cell sections4. Combined open and closed section5. Summary6. References


Qx

Qy

y

S

0

zsx

S

0

zs Qdssesens Qdssecosspp

z

x

y

O

z

x

y

Os

n

szs

Concept of Shear centre


z

x

y

O

z

x

y

O

MT

s

n

szs

T

S

0

zs Mdssrsesp

sr

A torsion moment is induced by transverse loads



szs

sr

Qx

Qy

O

Mz

zs s

Shear force, Qx

Shear force, Qy

Torsion moment, Mz



szs

sr

cx

cy Qx

Qy

O

CEC

cxcy

S

0

zs yQxQdssrsesp

cxcy

S

0

yQxQdssrsqp

by

ax

c

c

y xa Q b Q



Example

h

e

Qy

b

e

x

y

1 cm

10 cm

20 cm

e

b

h

x

y12

3 4


c


x

y12

3 4


q(s)

12

sI

heQsq

x

y

2

22

shsbhI

eQsq

x

y

3

2

2sb

I

heQsq

x

y

S1

S2

S3

c

Example



x

y12

3 4

c

x

ybS

I

bheQds

hsqdssrsq

p

42

2

22

11

00

𝑐 =𝑒 · ℎ2 · 𝑏2

4 · 𝐼𝑥= −3.75𝑐𝑚

𝑄𝑦 · 𝑒 · ℎ2 · 𝑏2

4 · 𝐼𝑥= −𝑄𝑦 · 𝑐

0

𝑆𝑝

𝑞 𝑠 · 𝑟 𝑠 · 𝑑𝑠 = −𝑄𝑦 · 𝑐Example


Alternative method to calculate Shear stresses in closed sections

The solution for a shear loaded closed section beam follows a similarpattern to that described in previous slides but with two importantdifferences.

• First, the shear flow includes the effect of shear force and the inducedtorsional moment (if shear force is not applied at shear centre)

• Secondly, the shear loads may be applied through points in the cross-section other than the shear centre so that torsional as well as sheareffects are included. This is possible since shear stresses produced bytorsion in closed section beams have exactly the same form as shearstresses produced by shear, unlike shear stresses due to shear andtorsion in open section beams.


Qy

Qx

In a beam subjected to transversal loads, both shear flow and normal stress appear

The equilibrium of a differentialelement in the z direction,neglecting body forces, yields to:

0

st

s

q z

z

zzz

z

The shear flow can be calculated including the torsional as well as shear effects in one expression

y

s

z

x

ss

ss

ss

qq

zz

qq

s zq

q

z

s

s

z

t



0

st

s

q z

Integrating as before

S

xyyx

xyxyyS

xyyx

xyyxxs dsty

PII

PQIQdstx

PII

PQIQqsq

02

020

0

02

02 s

S

xyyx

xyxyyS

xyyx

xyyxxqdssty

PII

PQIQdsstx

PII

PQIQsq

0sb qsqsq

Qy

Qx

y

s

z

x



Qy

Qx

We obtain qb by supposing that the closed beam section is cutat some convenient point thereby producing an open section

S

xyyx

xyxyyS

xyyx

xyyxxb dssty

PII

PQIQdsstx

PII

PQIQq

02

02

The value of shear flow at the cut (s = 0) is then found byequating applied and internal moments taken about someconvenient moment centre

dsspsqQQ oyox

dsspqdsspsqQQosboyox

2

dsspsqQQq

boyoxso

dssp2

is the area enclosed by the mid-line of the beam section wall

Qy

Qx

s

T

“cut”

So

A

s

qor

zo



Cell i Cell j Cell k

AB

C D

oiq

ojq

okq

oij

j

open qqqo

oj

j

open qq

oj

j

open qkqqo

oj

j

open qq

Multi-cell closed sections


Solution process for a section with n cells:

- The section in divided by n-cuts

- Each cell is solved as open

- Imposing in each cell:

j

openq

00

ds

pS

zs

0

dsG

qqqds

G

qq

dsG

qqqds

G

qq

A

D

kj

j

openD

C

j

j

open

C

B

ij

j

openB

A

j

j

open

ooo

ooo

System of n-equations with n-unknowns



0

dsG

qqqds

G

qq

dsG

qqqds

G

qq

A

D

kj

j

openD

C

j

j

open

C

B

ij

j

openB

A

j

j

open

ooo

ooo



Example

y

x

Qy = 1

0,5 cm

50 cm

12 cm

6 cm

e

L

h

R

1

2

L

h

e

R



y

x

y

x1

oq2

oq1

23

54

s2

s3

s6

s7

s4

s5

s1

*

0

*

s

ex

ex

a y

x

m y s e s ds

m sq s Q

I

+

Cuts at points 1 and 6

6


Example


y

x

1

23

54

s3

s6

s7

s4

s5

s1

6

s2

Skin 12

2* 1

2

1

2

2

ex

a

x

e sm

e sq s

I

Skin 23

2* 2

2

8 2

2 4

ex

a

x

h e se hm

e h hq s s

I

Skin 63

2* 6

2

6

2

2

ex

a

x

e sm

e sq s

I


Example


y

x

1

23

54

s3

s6

s7

s4

s5

s1

6

s2

Skin 15

2* 5

2

5

2

2

ex

a

x

e sm

e sq s

I

Skin 54

2* 4

4

8 2

2 4

ex

a

x

h e se hm

e h hq s s

I

Skin 64

2* 7

2

7

2

2

ex

a

x

e sm

e sq s

I


Example


y

x

1

23

54

s3

s6

s7

s4

s5

s1

6

s2

Skin 34

2 2* 2

0

22

cos4 2 4 2

1

4 2

ex

a

x

e h h e L e h h e Lm R e R d R e sen

e h h e Lq R e sen

I


Example


Shear flow in open section

y

x

9

xI

9

xI

9

xI

9

xI

69

xI

69

xI

78

xI

78

xI

96

xI


Example


y

x

1

23

54

Skin 12 12

1 1

o

aq s q s q

Skin 23 23

2 1

o

aq s q s q

Skin 63 63

6 2 1

o o

aq s q s q q

Skin 15 15

5 1

o

aq s q s q

Skin 54 54

4 1

o

aq s q s q

Skin 64 64

7 1 2

o o

aq s q s q q

Skin 34 34

2

o

aq s q q

1

oq2

oq

s3

s4

s5

s1

s2

s6

s7

6


Example


y

x

1

23

54

6

s3

s6

s7

s4

s5

s1

s2

0

0

pSq s

dse s

Imposing in each cell:

In cell 1

212 23

1 1 1 2 1 2

0 0

2 263 64

6 2 1 6 7 2 1 7

0 0

45 51

4 1 4 5 1 5

0 0

hL

o o

a a

h h

o o o o

a a

L L

o o

a a

q s q ds q s q ds

q s q q ds q s q q ds

q s q ds q s q ds


Example


y

x

1

23

54

6

s3

s6

s7

s4

s5

s1

s2

0

0

pSq s

dse s

234 64

2 7 1 2 2

0 0

263

6 1 2 6

0

0

h

o o o

a a

h

o o

a

q q R d q s q q ds

q s q q ds

1

2

37,58

77,44

o

x

o

x

qI

qI


Imposing in each cell:

In cell 2

Example


28,58

xI

23,86

xI

7,55

xI

31,42

xI

25,55

xI

23,86

xI

28,58

xI

7,55

xI31,42

xI

28,58

xI

28,58

xI

37,58

xI

9,528


Example


y

x

Qy = 100 kN

L

L/2

t

L/2

L/2

L/2

t = 2 mm

L = 200 mm

ycom = 125 mm

Ixx = 14.5·106 mm4

G=25000 N/mm2

ycom

Combined open and closed sections

com

Example


*

0

*

s

ex

ex

a y

x

m y s e s ds

m sq s Q

I

1

oq

s4

s3

s1

s9

s5

s2 s7

s6

s8

Cuts at point O

2

1

3

O

4 5

6 7

8

+


Shear force Qy


s4

s3

s1

s9

s5

s2 s7

s6

s82

1

3

O

4 5

6 7

8

Skin 23

Skin 12 Skin O3

mmNqL

q

ssdssI

Qsq

s

xx

y

/5.342

501069252

212

1

2

1

4

011112

1

mmNqL

q

sdsI

Qsq

s

xx

y

/5.1382

04.15.345.34752

23,323

20

2223

2


23,367,6

223767

qq

sqsq

27

112678

qq

sqsq

mmNqL

q

sdsI

Qsq

OO

s

xx

y

O

/1042

04.1752

3,33

30

333

3

3,36,6

3386

OO

OO

qq

sqsq

Shear force Qy


s4

s3

s1

s9

s5

s2 s7

s6

s82

1

3

O

4 5

6 7

8

Skin 34

mmNmmqq

mmNqLq

ssdssI

Qsq

s

xx

y

/7.28175

/5.172

106904.15.2425.242752

34max,34

434

2

4

4

40

44434

4


45

434956

qq

sqsq

mmNqqqq O /5.2420 3,323,334,334

Skin 45

mmNqLq

Lq

sdsI

Qsq

s

xx

y

/5.172

02

725.15.1725.1721252

545

45

50

5545

5

Shear force Qy



138.5 N/mm

104 N/mm

34.5 N/mm

242.5 N/mm

281.7 N/mm

172.5 N/mm

00

1 q

0

1

0

0

0

00

1

S

S

open

S

Sopen

ds

dssq

dsseG

dsseG

sq

q

Shear force Qy


z

x

y

O

Shear stresses

xxyy QKQK0qsq

Summary

2xyyx

x*eyxy

*ex

x

2xyyx

xy*eyy

*ex

y

PII

ImPmK

PII

PmImK

s

0A

*ey

s

0A

*ex

dssesxdAsxm

dssesydAsym

*

*


Single cell closed section

0qsqsq open

S

Sopen

dsseG

dsseG

sq

q

0

0

10

Summary


0

dsG

qqqds

G

qq

dsG

qqqds

G

qq

A

D

kj

j

openD

C

j

j

open

C

B

ij

j

openB

A

j

j

open

ooo

ooo

Multi-cell section

Summary


1. Megson, T.H.G. “Aircraft Structures”

Elsevier. 2007

Cap.16 Bending of open and closed thin-walled beams

Cap.17 Shear of beams

2. Benham, P.P. Y Crawford. “Mechanics of engineering materials”

Logman Scientific & Technical, 1987

Cap.5 Torsion

References


1. Cervera Ruiz, M. y Blanco Diaz, E. “Resistencia de Materiales”

Ed. UPC, 2001

Cap.7 Momento torsor

2. Garrido García, J.A. Y Foces Mediavilla, A. “Resistencia de Materiales”

Universidad de Valladolid, 1994

Cap.6 Torsión uniforme en barras de sección de pared delgada

3. Ortiz Berrocal, L “Elasticidad”

Ed. McGraw Hill, 1998

Cap.7 Torsión

4. Samartin Quiroga, A. “Elasticidad”

Ed. Bellisco. 1990

Cap.7 Estudio de la torsión

References

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