pure bending - chulalongkorn university: faculties and...
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1
Lect
ure
Not
e 5
Ben
ding
of T
hin
Plat
es
Firs
t Sem
este
r, A
cade
mic
Yea
r 201
2D
epar
tmen
t of M
echa
nica
l Eng
inee
ring
Chu
lalo
ngko
rn U
nive
rsity
Obj
ectiv
es
A
naly
ze b
endi
ng o
f thi
n pl
ates
A
naly
ze fo
r stre
ss a
nd d
efor
mat
ion
in p
late
s su
bjec
ted
to
bend
ing
and
twis
ting
bend
ing
and
twis
ting
A
naly
ze fo
r stre
ss a
nd d
efor
mat
ion
in p
late
s su
bjec
ted
to
dist
ribut
ed tr
ansv
erse
load
s
Ana
lyze
for s
tress
and
def
orm
atio
n in
pla
tes
subj
ecte
d to
co
mbi
ned
load
s
Des
crib
e en
ergy
met
hod
for a
naly
zing
thin
pla
tes
2
Topi
cs
Th
in p
late
s su
bjec
ted
to b
endi
ng, t
wis
ting,
dis
tribu
ted
trans
vers
e an
d in
-pla
ne lo
adin
g
Ben
ding
ofth
inpl
ates
with
asm
alli
nitia
lcur
vatu
re
Ben
ding
of t
hin
plat
es w
ith a
sm
all i
nitia
l cur
vatu
re
Ene
rgy
met
hod
C
hara
cter
istic
sS
ht
fthi
ti
l
3
S
heet
s of
thin
mat
eria
ls
Res
ist m
embr
ane
forc
es a
nd b
endi
ng
Pure
Ben
ding
Des
crip
tions
Th
ene
utra
lpla
nedo
esno
tdef
orm
and
isus
edas
the
4
Th
e ne
utra
l pla
nedo
es n
ot d
efor
m a
nd is
use
d as
the
refe
renc
e pl
ane.
P
lane
s re
mai
n pl
anes
Stra
ins
can
be d
eter
min
ed in
term
s of
z a
nd th
e ra
dii o
f cu
rvat
ure
r.
Dire
ct s
tress
es v
ary
linea
rly a
cros
s th
e th
ickn
ess.
2
Pure
Ben
ding
Str
ain
5
1(
xx
yz
xzE
2
1)
()
11
(
xx
y
yy
xz
y
Ez
zE
2
1)
()
1y
yx
Ez
Pure
Ben
ding
App
lied
boun
dary
Mom
ents
/2
2/2 /2
2/2
11
()
()
11
1(
)(
)1
t
xx
xt
xy
xy
t
yy
yt
Ez
My
zydz
Mdz
D
Ez
Mx
zxdz
Mdz
D
6
1y
xy
x
3
2Fl
exur
al ri
gidi
ty
12(1
)Et
D
1(
)
1(
)
xx
y
yy
x
MD
MD
Pure
Ben
ding
Def
lect
ion
22
2
22
2
22
2
22
2
11
()
()
11
()
()
xx
xy
y
ww
wM
DD
xx
y
ww
wM
DD
yy
x
7
22
2
22
22
If 0
:
11
1If
:(1
)
yy
yx
y xy
xy
yy
x
ww
My
xM
MM
MD
Ben
ding
& T
wis
ting
Des
crip
tion
8
3
Ben
ding
& T
wis
ting
Equi
libriu
m
0
cos
sin
sin
cos
nn
xy
xyxy
FMAC
MAB
MBC
MAB
MBC
9
22
co
ssi
nsi
n2
0
sin
cos
cos
sin
sin
2co
s22
0 0
nx
yxy
tt
xy
xyxy
x
n
yt
txy
MM
MM
FMAC
MAB
MBC
MAB
F F
MBC
MM
MM
Ben
ding
& T
wis
ting
Prin
cipa
l Mom
ents
sin
2co
s22
0
Prin
cipa
l mom
ents
and
cur
vatu
res
xy
txy
t
MM
MM
M
10
2ta
n2
xy
xy
MM
M
Ben
ding
& T
wis
ting
Shea
r #1
xyxy
My
yzdz
Mx
xzdz
11
/2 /2
/2 /2
xyxy
t
xy x
xyt
t
yxy
t
Mx
xzdz
Mzdz
Gzdz
M
Ben
ding
& T
wis
ting
Shea
r #2
,
ww
uzv
zx
yv
u
2
2
x xyyx
y wzxy
12
23
2/2
/22
/2/2
32
2
26
(1)
12(1
)
tt
xyxy
tt
wGt
wM
Gzdz
Gz
dzxy
xy
Et
ww
Dxy
xy
4
Tran
sver
se L
oad
Des
crip
tion
13
She
ar s
train
s ar
e ig
nore
d.
Tran
sver
se L
oad
Equi
libriu
m #
1
/2 /2 /2 /2 0
t
xxz
t t
yyz
t
Qdz
Qdz
F
14
0
()
()
0
0
z
yx
xx
yy
yxF
Qxy
Qy
Qy
xQ
xqxy
xy
qx
y
Tran
sver
se L
oad
Equi
libriu
m #
2
/2 /2 /2 /2
t
xx
t t
yy
t
Mzdz
Mxdz
/2 /2
22
2
0
()
()
t
xyyx
xyt
x
xyy
xyxy
yy
MM
xdz
M
MM
Mxy
My
My
xM
xx
yQ
Qy
yy
15
()
()
02
22
0 0
yx
yx
x
xyy
y
xyx
x
yy
yQ
yxy
Qx
Qqx
yx
MM
Qx
yM
MQ
yx
Tran
sver
se L
oad
Equi
libriu
m #
3
22
20
0xy
yxy
yy
y
MM
MM
xy
yx
yy
22
2
22
22
00
From
xyxy
xx
xx y
x
xyy
xyx
xy
yx
yy
MM
MM
yx
xy
xx
qx
yM
MM
M
16
22
22
2
22
2
xyy
xyx
xyy
x
qxy
yx
xy
MM
Mq
xy
xy
5
Tran
sver
se L
oad
Dis
plac
emen
t #1
22
22
2
22
22
(),
(
),
(1)
xy
xyw
ww
ww
MD
MD
MD
xy
xy
yx
22
2
22
22
22
22
22
22
22
22
42
4
From
2
()
2(1
)(
)
()
2(1
)
xyy
x
yy
yM
MM
qxy
xy
ww
ww
wD
DD
qxy
xy
xx
yy
yx
ww
w
4
4
()
ww
q
17
42
22
()
2(1
)x
xy
x
24
22
44
4
42
24
()
2
Dy
yxy
ww
wq D
xxy
y
22
22
2(
)q
wD
xy
Tran
sver
se L
oad
Dis
plac
emen
t #2
2
2M
Mw
w
22
22
22
()
()
xyx
x
yxy
y
MM
ww
QD
xy
xx
yM
Mw
wQ
Dy
xy
xy
18
Tran
sver
se L
oad
BC
#1:
Sim
ple
Supp
ort
Free
to ro
tate
, no
defle
ctio
nA
long
aned
gex
2
2
22
22
Alo
ng a
n e
dge
0,
0
M(
)0
BC BC
x ww
wy
yw
wD
xy
19
2
20,
()
0BC
BC
ww
x
Tran
sver
se L
oad
BC
#2:
Bui
lt-in
No
toro
tatio
n,no
defle
ctio
nN
o to
rota
tion,
no
defle
ctio
nA
long
an
edg
ex
0,
0
BC
ww
x
20
6
Tran
sver
se L
oad
BC
#3:
Fre
e
No
bend
ing
mom
ents
, tw
istin
g m
omen
ts o
r ver
tical
she
arin
g fo
rces
Alo
ng a
n e
dge
()
0(
)0
()
0x
MM
Q
33
32
22
()
0, (
)0,
()
0
()
0
((2
))
0
xBC
xyBC
xBC
xyx
BC
BC
MM
Q
MQ
yw
wx
xy
ww
21
22
()
()
0xBC
BC
ww
Mx
y
Tran
sver
se L
oad
BC
#3:
Fre
e
No
bend
ing
mom
ents
, tw
istin
g m
omen
ts o
r ver
tical
she
arin
g fo
rces
Alo
ng a
n e
dge
()
0(
)0
()
0x
MM
Q
33
32
22
()
0, (
)0,
()
0
()
0
((2
))
0
xBC
xyBC
xBC
xyx
BC
BC
MM
Q
MQ
yw
wx
xy
ww
22
22
()
()
0xBC
BC
ww
Mx
y
Exam
ple
Tran
sver
se L
oad
#1
A si
mpl
y su
ppor
ted
plat
e of
dim
ensi
on a
×b
is s
ubje
cted
to a
un
iform
tran
sver
se lo
ad q
. Det
erm
ine
the
defle
ctio
n an
d be
ndin
g m
omen
t dis
tribu
tions
.
23
Exam
ple
Tran
sver
se L
oad
#2
44
4
42
24
The
defle
ctio
n m
ust s
atis
fy
2w
ww
q Dx
xy
y
2
2
2
2
Bou
ndar
y co
nditi
ons
0 an
d a
t 0
and
0 an
d a
t y0
and
y
Dx
xy
y
ww
xx
ax w
wb
y
24
11
11
Thus
sin
sin
sin
sin
mn
mn
mn
mn
mx
ny
wA
ab
mx
ny
qa
ab
7
Exam
ple
Tran
sver
se L
oad
#3
00
sin
sin
ab
ab
mx
ny
qdxdy
ab
mx
ny
mx
ny
00
11
0
sin
sin
sin
sin
4
For
sin
sin
ab
mn
mn
mn
a
mx
ny
mx
ny
adxdy
ab
ab
aba
mx
mx
fdx
aa
a
25
00 w
hen
and
w
hen
2
For
sin
sin
0 w
hen
and
w
he2
b
af
mm
fm
m
ny
ny
gdy
bb
bg
nn
f
n n
n
Exam
ple
Tran
sver
se L
oad
#4
44
4
42
24
20
ww
wq D
xxy
y
42
24
24
22
()
2()
()
()
0
()
()
0
1
mn
mn
mn
mn
am
mn
nA
aa
bb
D
am
nA
ab
D
26
42
22
11
11
1si
nsi
n(
/)
(/
)
sin
sinmn
mn
mn
mn
amx
ny
wa
bD
ma
nb
mx
ny
qa
ab
Exam
ple
Tran
sver
se L
oad
#5
20
0
416
sin
sin
ab
mn
qmx
ny
qa
dxdy
aba
bmn
62
22
1,3,5
1,3,
5
max
62
22
1,3,
51,3
,5
16si
n(/
)sin
(/
)
(/
)(
/)
16si
n(/2
)sin
(/2
)at
/2
,/2
(/
)(
/)
mn
mn
aba
bmn
qmxa
nyb
wD
mnma
nb
qm
nw
xa
yb
Dmnma
nb
27
,,
,,
max
(/
)(
/)
0.04
43
mnma
nb
wq
4 3
if
,0.
3a
ab
Et
Exam
ple
Tran
sver
se L
oad
#62
2
42
22
1,3,
51,
3,5
22
(/
)(
/)
16si
nsi
n(
/)
(/
)
(/
)(
/)
xm
n
ma
nb
qmx
ny
Ma
bmnma
nb b
22
42
22
1,3,5
1,3,
5
,max
,max
,max
,max
(/
)(
/)
16si
nsi
n(
/)
(/
)
at
/2,
/2
0
ym
n
xy
xy
ma
nb
qm
xny
Ma
bmnma
nb
MM
xa
yb
MM
2.0
479
if
,0.
312
12y
x
qaa
b
Mz
Mz
28
33
,max
,max
22
2
,max
,max
2
,
66
, a
t 2
0.28
71 if
,
0.3
yx
xy
yx
xy
xy
ttM
Mt
zt
t aq
ab
t
8
Com
bine
d Lo
adin
gs D
escr
iptio
ns
Com
bine
d lo
ads
Tl
d
Tran
sver
se lo
ads
In
-pla
ne fo
rces
29
Com
bine
d Lo
adin
gs E
quili
briu
m #
1
0yx
xN
N xy
0xy
yN
Nx
y
30
2
20
()
cos(
)co
s()
()
0
xx
xx
yxyx
yx
Nw
ww
FN
xy
xN
yx
xx
xN
Ny
xN
xy
Com
bine
d Lo
adin
gs E
quili
briu
m #
2
2xy
Nw
ww
31
2 2
()
()
()
()
((
))xy
z
xyxy
zxy
xy
xyxy
yxyx
zyx
Nw
ww
NN
xy
xN
yx
xxy
yN
ww
Nxy
xy
xy
xy
Nw
wN
Nxy
xy
xy
yx
N
Com
bine
d Lo
adin
gs E
quili
briu
m #
3
2N
ww
w
32
2
2
2
2
2
()
()
()
()
()
()
xxz
xx
xx
yyz
y
xz
Nw
ww
NN
xy
xN
yx
xx
xN
ww
Nxy
xy
xx
xN
ww
NN
xy
xy
yy
N
y
9
Com
bine
d Lo
adin
gs E
quili
briu
m #
4
()
()
()
()
zz
zxy
zyx
zxz
yz
RF
RN
NN
N
2
22
22
22
2
22
)2
(2
)
xyxy
zxy
yx
xy
zx
yxy
z
NN
ww
wR
xy
Nxy
xy
xy
xy
yx
NN
ww
ww
Nxy
xy
Nxy
xy
xx
yy
xy
ww
wR
NN
Nxy
xy
xy
R
33
from
yx
x
xy
xy
NN x
y
0
and
0 an
d in
tran
sver
se lo
adxy
yN
Nx
y
44
42
22
42
24
22
12
(2
)x
yxy
ww
ww
ww
qN
NN
Dxy
xxy
yx
y
Initi
al C
urva
ture
Des
crip
tions
01
04
44
11
14
22
4
Ass
ume
tota
l def
lect
ion
as
the
sum
e of
initi
al
2
ww
ww
ww
w
42
24
22
20
10
10
12
2
2
0
()
()
()
1(
22)
Initi
al c
urva
ture
is e
quiv
alen
t to
the
appl
icat
ion
of
xy
xy
x
xxy
yw
ww
ww
wq
NN
ND
xy
xy
qN
22
00
02
22
yxy
ww
wN
Nxy
xy
34
01
1
12
62
22
21
1
sin
sin
, if
is c
ompr
essi
ve,
0
sin
sin
, (
/)
(/
)
mn
xy
xym
n
mn
xmn
mn
mn
x
xy
xy
mx
ny
wA
NN
Na
bAN
mx
ny
wB
Ba
bDa
mna
mb
N
Ener
gy M
etho
d D
escr
iptio
ns
35
22
2
22
22
22
2
22
22
1(
2)
2
From
(
),(
),(1
)
xy
xy
xy
xy
ww
wU
MM
Mxy
xy
xy
ww
ww
wM
DM
DM
Dxy
xy
yx
Ener
gy M
etho
d B
endi
ng a
nd T
wis
ting
22
22
22
22
22
22
22
22
22
2
()
()
22(
1)(
)2
For a
re
ctan
gula
r pla
tea
b
Dw
ww
ww
Uxy
xy
xy
xy
ab
Dw
ww
ww
36
22
22
22
00
2
2
((
)2(
1)
()
2
For b
endi
ng o
nly,
0
(2
ab
xy
Dw
ww
ww
Udxdy
xy
xy
xy
M
Dw
Ux
22
22
22
20
0(
)2
ab
ww
wdxdy
yx
y
10
Ener
gy M
etho
d Tr
ansv
erse
Loa
d
00
()
ab
Vwqxy
Vwqdxdy
37
Ener
gy M
etho
d In
-pla
ne L
oads
#1
22
21
(1(
))
2
ax
ww
xx
a
38
2
0
2
0
2 1(1
()
)2 1
()
2(
)
a
a xx
xw
adx
x wa
adx
xV
Ny
Na
ay
Ener
gy M
etho
d In
-pla
ne L
oads
#2
21
()
ab
wV
Ndxdy
39
2
00
2
00
()
2 1(
)2
xx
ab
yy
VN
dxdy
x wV
Ndxdy
y
Ener
gy M
etho
d In
-pla
ne L
oads
#3
1 2 1 2
xyxy xy
xy
ww
VN
xx
yw
wN
yV
40
00
22
00
0
2 12
21
()
()
22
xy ab
xyxy
ab
b
xy
xyx
y
x
xy
yyx
y ww
VN
dxdy
xy
ww
ww
VV
VV
NN
Ndxdy
xy
xy
11
Exam
ple
Ener
gy #
1
A si
mpl
y su
ppor
ted
plat
e of
dim
ensi
on a
×b
is s
ubje
cted
to a
un
iform
tran
sver
se lo
ad q
. Det
erm
ine
the
defle
ctio
n an
d be
ndin
g m
omen
t dis
tribu
tions
.
1
22
22
22
22
22
20
0
sin
sin
((
)2(
1)
()
2mn
m
abm
xny
wA
ab
Dw
ww
ww
UV
dxdy
xy
xy
xy
41
00
62
22
1,3,
51,
3,5
2
()
16si
n(/
)sin
(/
)
(/
)(
/)
ab
mn
xy
xy
xy
wqdxdy
qmxa
nyb
wD
mnm
U
anb
V