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

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

Aerospace Structures

Chapter 2. Bending, shear and torsion of thin-walled beams Torsion on thin-walled beams


OBJETIVES

Objetives

• Knowledge of the hypothesis of torsion in thin-walled beams

• Calculate the shear stress distribution on open sections

• Calculate the shear stress distribution on single-cell closed sections

• Calculate the shear stress distribution on multiple cell sections

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

Torsion on thin walled beams


Index

1. Introduction2. Stress field on open sections3. Stress field on single-cell closed sections4. Stress field on multiple cell sections5. Summary6. References

Index




Introduction

1. Introduction2. Stress field on open sections3. Stress field on single-cell closed sections4. Stress field on multiple cell sections5. Summary6. References

Index




y,xwz,y,xu

zxz,y,xu

zyz,y,xu

z

zy

zx

Kinematic hypothesis

Displacements field

Stresses field

02 , ,w x y x y

x

y

yz

xz

G22

dAy,x2M

A

T

Introduction

Previous knowledge


Uniform torsionThe beam is subjected to a uniform torsional moment while sectiondisplacements are not constrained by boundary conditions

Saint-Venant hypothesis

Displacement field is produced by:

- A rotation of each section about a point. This point is the torsion centre

- Every section experience the same warping displacement

Introduction


Cross-section of a thin-walled beam

H

B

e1

e2

B,HMin10

1 e , e 21

OPEN

CLOSED

Single cell

Multi cell

Introduction


1. Introduction2. Stress field on open sections3. Stress field on single-cell closed sections4. Stress field on multi-cell sections5. Summary6. References

Index

Stress field on open sections


Session 4: Torsion on thin walled beams


A

B

e

x

z

y

Torsion function

Torsion function is independent oncoordinate s, except in extreme points

x

y

G2

x2

2

e

b

MT

s

n

z

Single branch section


0x 2

ex en

G22

xy,x


x

y

e

b

MT

4

exGx

22

0

2

xz

yz G x

dAx2M

A

T

3T ebG

3

1M

3eb3

1J




JGMT

3

3

1ebJ



J

nMTsz

2

A

B

e

s

n

z

J

Me Tsz

max,

pS

dsseJ0

3

3

1

Uniform thickness: Non-uniform thickness:


x

y

Shear stress xz

is acting on this regionyz

xz

xz




yz

Calculating the resultant force of shearstresses:

32

e

2e

yz ebG6

1dxbx

3T ebG

3

1M

Shear stress yz is in equilibrium with MT/2

Since:

xz equilibrates the other MT/2




x

y

iiiT JGM

3iii eb

3

1J

xG2 iiyz

For each branch

yz

xi

yi

Hypothesis: The stress concentrationat the corners is neglected i

iiTT MM

3i

iglobal eb

3

1J

global

T

i

T

n

T

2

T

1

T

JG

M

JG

M

JG

M

JG

M

JG

Min21


Branched sections



Index

Stress field on closed sections




Comparison between open and closed sections

Linear shear stresses

Uniform shear stresses



x

y

MT

1zs s

2zs s

1ss 2ss 1zs s

2zs s

1se

2se

dzsesdzses 22zs11zs

22zs2T

11zs1T

sessq

sessq

ctesqT



x

y

MT

1ss 2ss 1zs s

2zs s

dssrsesM zsT

2qM TT

dssr2

2dsszs

se

ds

4GM

2

T

se

ds

4J

2

Using circular

sector concept



x

y

MT

0s pSs

dssrz

ds0usus

0

zs

0

zszz

dssrz

ds0uSsupp Ss

0

z

Ss

0

zszpz

pzz Ssusu

0

1

2

s

o s r s ds

0

0

0 2

0 2

p

s

zz z zs o

s S

zz p z zs o p

u s u ds sz

u s S u ds s Sz

Using circular

sector concept



x

y

MT

0s pSs

0 0

p ps S s Ss

s

pzz Ssusu

sSsz

2ds0 opoz

Ss

s

zs

p

z2ds z

S

0

zs

p

0 0

p ps S s S s

s

o p os S s


se

Ms T

2

min

max2 e

MT


Closed sections are more effective to resisttorsional moments

Example: circular sections

x

y

MT

x

y

MT

e

ds

RJ

eRJ

C

O

22

3

4

23

1

eRJ

eRJ

C

O

3

3

2

3

2

13

12

R

e

J

J

C

O1

3

12

R

e

O

C




x

y

MT

x

y

MT

e

M

J

eM

TC

O

TO

2

max

13

max

e

R

C

O

eR

M

eR

M

TC

TO

2

2max

2

2

3



Closed sections are more effective to resisttorsional moments

Example: circular sections



Index

Multi-cell closed sections





TORSIONAL MOMENT

Cell 1 Cell 2 Cell 3

C

2,mq1,mq3,mq

1,mq

1,mq

2,mq

2,mq3,mq

3,mq

A B D

E F G H

Compatibility equations:

𝛿𝑖𝑖 =1

𝐺

𝑖

𝑑𝑠

𝑡(𝑠)

𝛿𝑖𝑗 = −1

𝐺

𝑖𝑗

𝑑𝑠

𝑡(𝑠)

𝛿11 · 𝑞𝑚,1 + 𝛿12 · 𝑞𝑚,2 − 2 · Ω1 · 𝑤 = 0

𝛿22 · 𝑞𝑚,2 + 𝛿12 · 𝑞𝑚,1 + 𝛿23 · 𝑞𝑚31 − 2 · Ω2 · 𝑤 = 0

𝛿33 · 𝑞𝑚,3 + 𝛿23 · 𝑞𝑚,2 − 2 · Ω3 · 𝑤 = 0

𝑀𝑇 =

𝑖

2 · Ω𝑖 · 𝑞𝑖


100mm

1mm

Example

2mm

1mm4mm

1mm1mm3mm

100mm 100mm

MT

MT= 5 kNmG= 25000 MPa

Find the maximum shear stress

max= 36 MPa



Combined open and closed sections

2

LQM yT

MT

opcl

T

JJJ

JGM

4646

433

4622

101610001.16

7.106621003

14

3

1

10164200

220044

mmmmJJJ

mmtsJ

mm

st

dsJ

opcl

op

cl

mmradGJ

MT /1025101625000

10010100 6

6

3

MPat

q

mmNJG

q

clcl

clcl

5.62

/1252

Closed cell

MPaGtop 25.11025250002 6

max,

Open branches

Maximum shear stress (Qy +MT): 344.2 MPa




Index

Summary




JGMT3eb

3

1J

eGmax

Single branched open sections

Closed sections

e

MT

2

se

ds

4J

2 JGMTx

y

x

y

Summary


Multiple branched sections

iiiT JGM

3iii eb

3

1J

xG2 iiyz

i

iiTT MM

3i

iglobal eb

3

1J

x

y

Summary



Index

References




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

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

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