chapter 2. bending, shear and torsion of thin-walled beams
TRANSCRIPT
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
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
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
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
Index
1. Introduction2. Stress field on open sections3. Stress field on single-cell closed sections4. Stress field on multiple cell sections5. Summary6. References
Index
CHAPTER 2. Bending, shear and torsion of thin-walled beams
Torsion on thin walled beams
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
Introduction
1. Introduction2. Stress field on open sections3. Stress field on single-cell closed sections4. Stress field on multiple cell sections5. Summary6. References
Index
CHAPTER 2. Bending, shear and torsion of thin-walled beams
Torsion on thin walled beams
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
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
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
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
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
Cross-section of a thin-walled beam
H
B
e1
e2
B,HMin10
1 e , e 21
OPEN
CLOSED
Single cell
Multi cell
Introduction
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
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
CHAPTER 2. Bending, shear and torsion of thin-walled beams
Session 4: Torsion on thin walled beams
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
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
Stress field on open sections
0x 2
ex en
G22
xy,x
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
x
y
e
b
MT
4
exGx
22
0
2
xz
yz G x
dAx2M
A
T
3T ebG
3
1M
3eb3
1J
Stress field on open sections
Single branch section
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
JGMT
3
3
1ebJ
Stress field on open sections
Single branch section
J
nMTsz
2
A
B
e
s
n
z
J
Me Tsz
max,
pS
dsseJ0
3
3
1
Uniform thickness: Non-uniform thickness:
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
x
y
Shear stress xz
is acting on this regionyz
xz
xz
Stress field on open sections
Single branch section
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
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
Stress field on open sections
Single branch section
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
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
Stress field on open sections
Branched sections
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
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 closed sections
CHAPTER 2. Bending, shear and torsion of thin-walled beams
Torsion on thin walled beams
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
Comparison between open and closed sections
Linear shear stresses
Uniform shear stresses
Stress field on closed sections
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
x
y
MT
1zs s
2zs s
1ss 2ss 1zs s
2zs s
1se
2se
dzsesdzses 22zs11zs
22zs2T
11zs1T
sessq
sessq
ctesqT
Stress field on closed sections
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
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
Stress field on closed sections
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
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
Stress field on closed sections
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
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
Stress field on closed sections
se
Ms T
2
min
max2 e
MT
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
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
Stress field on closed sections
Comparison between open and closed sections
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
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
Stress field on closed sections
Comparison between open and closed sections
Closed sections are more effective to resisttorsional moments
Example: circular sections
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
1. Introduction2. Stress field on open sections3. Stress field on single-cell closed sections4. Stress field on multi-cell sections5. Summary6. References
Index
Multi-cell closed sections
CHAPTER 2. Bending, shear and torsion of thin-walled beams
Torsion on thin walled beams
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
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
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
100mm
1mm
Example
2mm
1mm4mm
1mm1mm3mm
100mm 100mm
MT
MT= 5 kNmG= 25000 MPa
Find the maximum shear stress
max= 36 MPa
Multi-cell closed sections
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
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
Multi-cell closed sections
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
1. Introduction2. Stress field on open sections3. Stress field on single-cell closed sections4. Stress field on multi-cell sections5. Summary6. References
Index
Summary
CHAPTER 2. Bending, shear and torsion of thin-walled beams
Torsion on thin walled beams
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
JGMT3eb
3
1J
eGmax
Single branched open sections
Closed sections
e
MT
2
se
ds
4J
2 JGMTx
y
x
y
Summary
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
Multiple branched sections
iiiT JGM
3iii eb
3
1J
xG2 iiyz
i
iiTT MM
3i
iglobal eb
3
1J
x
y
Summary
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
1. Introduction2. Stress field on open sections3. Stress field on single-cell closed sections4. Stress field on multi-cell sections5. Summary6. References
Index
References
CHAPTER 2. Bending, shear and torsion of thin-walled beams
Torsion on thin walled beams
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
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
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
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