Department of Continuum Mechanics and Structural...
Transcript of 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)
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
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)
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
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
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
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
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
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
Concept of Shear centre
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
szs
sr
Qx
Qy
O
Mz
zs s
Shear force, Qx
Shear force, Qy
Torsion moment, Mz
Concept of Shear centre
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
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
Concept of Shear centre
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
Example
h
e
Qy
b
e
x
y
1 cm
10 cm
20 cm
e
b
h
x
y12
3 4
Concept of Shear centre
c
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
x
y12
3 4
Concept of Shear centre
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
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
Concept of Shear centre
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
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
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.
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
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
zz
s zq
q
z
s
s
z
t
Alternative method to calculate Shear stresses in closed sections
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
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
Alternative method to calculate Shear stresses in closed sections
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
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
Alternative method to calculate Shear stresses in closed sections
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
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
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
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
dsG
qqqds
G
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
Multi-cell closed sections
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
0
dsG
qqqds
G
dsG
qqqds
G
A
D
kj
j
openD
C
j
j
open
C
B
ij
j
openB
A
j
j
open
ooo
ooo
Multi-cell closed sections
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
Example
y
x
Qy = 1
0,5 cm
50 cm
12 cm
6 cm
e
L
h
R
1
2
L
h
e
R
Multi-cell closed sections
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
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
Multi-cell closed sections
Example
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
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
Multi-cell closed sections
Example
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
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
Multi-cell closed sections
Example
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
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
Multi-cell closed sections
Example
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
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
Multi-cell closed sections
Example
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
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
Multi-cell closed sections
Example
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
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
Multi-cell closed sections
Example
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
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
Multi-cell closed sections
Imposing in each cell:
In cell 2
Example
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
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
Multi-cell closed sections
Example
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
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
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
*
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
+
Combined open and closed sections
Shear force Qy
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
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
Combined open and closed sections
23,367,6
223767
sqsq
27
112678
sqsq
mmNqL
q
sdsI
Qsq
OO
s
xx
y
O
/1042
04.1752
3,33
30
333
3
3,36,6
3386
OO
OO
sqsq
Shear force Qy
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
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
Combined open and closed sections
45
434956
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
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
Combined open and closed sections
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
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
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
*
*
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
Single cell closed section
0qsqsq open
S
Sopen
dsseG
dsseG
sq
q
0
0
10
Summary
Authors: Enrique Barbero Pozuelo, José Fernández Sáez, Carlos Santiuste Romero
0
dsG
qqqds
G
dsG
qqqds
G
A
D
kj
j
openD
C
j
j
open
C
B
ij
j
openB
A
j
j
open
ooo
ooo
Multi-cell section
Summary
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