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1
Lect
ure
Not
e 4
Virt
ual W
ork
& E
nerg
y M
etho
d
Seco
nd S
emes
ter,
Aca
dem
ic Y
ear 2
012
Dep
artm
ent o
f Mec
hani
cal E
ngin
eerin
gC
hula
long
korn
Uni
vers
ity
Obj
ectiv
es
U
se th
e en
ergy
met
hod
to a
naly
ze s
truct
ures
D
escr
ibe
the
char
acte
ristic
s an
d pr
oper
ties
as w
ell a
s de
term
ine
stra
inen
ergy
and
com
plem
enta
ryen
ergy
and
dete
rmin
e st
rain
ene
rgy
and
com
plem
enta
ry e
nerg
y an
d po
tent
ial e
nerg
y
Des
crib
e th
e pr
inci
ple
of v
irtua
l wor
k an
d us
e th
e pr
inci
ple
to d
eter
min
e eq
uilib
rium
, sta
bilit
y an
d an
alyz
e si
mpl
e el
astic
ity p
robl
ems
with
em
phas
is o
n be
ndin
g pr
oble
ms
A
sim
ple
stat
ical
ly in
dete
rmin
ate
prob
lem
s w
ith e
mph
asis
on
ben
ding
2
Topi
cs
V
irtua
l Wor
k
Stra
in e
nerg
y, c
ompl
emen
tary
and
pot
entia
l ene
rgy
Def
lect
ions
D
efle
ctio
ns
Sta
tical
ly in
dete
rmin
ate
prob
lem
s
3
Wor
k B
y a
Forc
e
co
sF
WFdr
Fdr
4
wor
k
forc
e th
at d
one
the
wor
kdi
spla
cem
ent
FW
F dr
2
Wor
k B
y a
Cou
ple
()
()
()
22
Mr
rW
FF
Fr
5
22
MW
M
mag
nitu
de o
f cou
ple
that
do
the
wor
ksm
all a
ngle
of r
otat
ion
M
Virt
ual W
ork
Virt
ual M
ovem
ents
Im
agin
ary
or v
irtua
l mov
emen
tsis
ass
umed
and
doe
s no
t ac
tual
ly e
xist
.
Virt
ual d
ispl
acem
ent
V
irtua
l rot
atio
n
Virt
ual d
efor
mat
ion
V
irtua
l mov
emen
ts a
re in
finite
sim
ally
sm
all a
nd d
oes
not
viol
ate
phys
ical
con
stra
ints
.
6
Prin
cipl
e of
virt
ual w
ork
is a
n al
tern
ativ
e fo
rm o
f N
ewto
n’s
law
s th
at c
an a
naly
ze th
e sy
stem
in
equi
libriu
m u
nder
wor
k an
d en
ergy
con
cept
s.
Virt
ual W
ork
Prin
cipl
e of
Virt
ual W
ork
C
onsi
der a
n ob
ject
in e
quilib
rium
Th
e vi
rtual
wor
k do
ne b
y al
l for
ces
to m
ove
the
obje
ct w
ith
itld
il
ta
virtu
al d
ispl
acem
ent
1co
sr
Fk
vk
kW
F
Iilib
i0
W
7
In
equ
ilibriu
m,
0F
W
Virt
ual W
ork
Prin
cipl
e of
Virt
ual W
ork
for R
igid
Bod
ies
te
iW
WW
tota
l virt
ual w
ork
done
exte
rnal
wor
k do
nein
tern
al v
irtua
l wor
k
t e i
W W W
e
iW
W
8
3
Exer
cise
Virt
ual W
ork
for R
igid
Bod
ies
#1
Det
erm
ine
the
supp
ort r
eact
ions
,,
,,
00
0
vB
vC
t CvC
vB
a LW R
Wa
RW
9
,,
0C
vC
vC
C
RWL
aR
WL
Exer
cise
Virt
ual W
ork
for R
igid
Bod
ies
#2
10
0(
)(
)0
()
()
00
and
0
t Av
vv
Cv
v
AC
vC
v
AC
C
W RW
aR
LR
RW
RLWa
RR
WRLWa
Virt
ual W
ork
Virt
ual W
ork
for D
efor
mab
le B
odie
s
te
iW
WW
11
Virt
ual W
ork
Inte
rnal
Virt
ual W
ork
from
Axi
al L
oad
NN
AA
AN
,
, ,
,
()
iNv
A
v
iNv
L
i
i N
N
v
Nw
dAx
AN
x
wN
dx
wN
dx
w
12
L
vv
vN
EEA
,A
viN
L
NN
wdx
EA
4
Virt
ual W
ork
Inte
rnal
Virt
ual W
ork
from
Tor
sion
,A
viT
L
TT
wdx
GJ
13
Virt
ual W
ork
Inte
rnal
Virt
ual W
ork
from
Ben
ding
,A
viM
L
MM
wdx
EI
14
Virt
ual W
ork
Inte
rnal
Virt
ual W
ork
from
She
ar F
orce
S
A
, , ,
()
()
()
iSv
A
vA
v
iS iS
wdA
x
SdA
xA
Sx
wS
dx
w w
15
,
Av
iSL
SS
wdx
GA
,iSv
L
vv
vwS
dx
GS GA
Virt
ual W
ork
Virt
ual W
ork
from
Ext
erna
l Loa
ds
,
,e
vy
vx
wW
P
MT
,(
)V
ev
Le
vy
wM
T
wx
xw
d
16
,,
,(
()
)
()
evy
vx
Vv
vy
L
Av
Av
Av
Av
iA
vL
LL
L
WW
PM
Twx
dx
NN
SS
MM
TT
Wdx
dxdx
dxM
EA
GA
EI
GJ
5
Exer
cise
Virt
ual W
ork
for D
efor
mab
le B
odie
s #1
Det
erm
ine
the
bend
ing
mom
ent a
t B
,vB
ab
a b
17
Exer
cise
Virt
ual W
ork
for D
efor
mab
le B
odie
s #2
Det
erm
ine
the
bend
ing
mom
ent a
t B
,
(1)
00
B
t
vB
BB
aL
bb
W WM
18
,vB
BB
B
B
LWa
Mb
Wab
ML
Exer
cise
Virt
ual W
ork
for T
russ
#1
Det
erm
ine
the
forc
e in
AB
19
Exer
cise
Virt
ual W
ork
for T
russ
#2
,
,
,
34
43
030
0
vB
vB
C
t
CBA
vB
WF
20
40
kN
BA
F
6
Exer
cise
Virt
ual W
ork
for C
antil
ever
Bea
m #
1
Det
erm
ine
the
end
defle
ctio
n
21
Exer
cise
Virt
ual W
ork
for C
antil
ever
Bea
m #
2
2
()
wM
Lx
,
3,
()
2 1()
1(1
)
()
2
A v iMB
Av
iML
L
L
ML
x
ML
xW
v MM
wW
dxL
xdx
EI
EI
w
22
4
,0
4
,
()
8
From
(1),
18L
iM
BiM
wW
Lx
EI
wL
vW
EI
Stra
in E
nerg
y D
efin
ition
S
train
ene
rgy U
: ene
rgy
stor
ed in
mem
ber
C
ompl
emen
tary
ene
rgy
C: n
o ph
ysic
al m
eani
ng b
ut o
beys
the
law
ofen
ergy
cons
erva
tion
Ener
gy
law
of e
nerg
y co
nser
vatio
n
23
0y
UPdy
0P
CydP
Stra
in E
nerg
y R
elat
ions
hips
Ener
gy
, dU
dCP
ydy
dP
1/
00
00
Ass
umin
g fu
nctio
n 1
()
n
yP
n
Py
n
dydP
Pby
PU
Pdy
dPn
b
CydP
nby
dy
24
D
eter
min
e a
nd
for
linea
r ela
stic
mat
eria
l U
C
7
Com
plem
enta
ry E
nerg
y Pr
inci
ple
Ener
gy
For a
n el
astic
bod
y in
equ
ilibr
ium
und
er th
e ac
tion
of
appl
ied
forc
esth
etr
uein
tern
alfo
rces
(ors
tres
ses)
and
appl
ied
forc
es, t
he tr
ue in
tern
al fo
rces
(or s
tres
ses)
and
reac
tions
are
thos
e fo
r whi
ch th
e to
tal c
ompl
emen
tary
en
ergy
has
a s
tatio
nary
val
ue.
Com
patib
ility
0
n
WW
WydP
P
25
1
01
0
()
()
0
te
ir
rV
r nP
ie
rr
Vr
WW
WydP
P
CC
ydP
P
Exam
ple
Def
lect
ion
#1En
ergy
Det
erm
ine
the
defle
ctio
n,
cros
s se
ctio
nal a
rea A
= 18
00 m
m2 ,
E=
200
GPa
.
26
21
22
0k
ii
i
ii
i
FL
FC P
AE
P
Exam
ple
Def
lect
ion
#2En
ergy
Rea
l loa
dIm
agin
ary
load
27
Exam
ple
Def
lect
ion
#3En
ergy
6
112
6810
Nm
m3
52k
iFFL
,2
52
1
6
,2
52
1
3.52
mm
(180
0 m
m)(
210
N/m
m)
188
010
Nm
m2.
44 m
m(1
800
mm
)(2
10 N
/mm
)
iBv
ii
i ki
Dh
ii
i
FL
AE
P FFL
AE
P
28
8
Exam
ple
SI P
robl
em #
1En
ergy
Red
unda
nt
mem
ber
0
1
ik
F
ii
ik
CdF
P
dFC
1
1
0
11
0(4
.83
2.70
7)
0
0.56
ii
ik
iii
i
dFC R
R FFL
RL
PL
AE
RAE
RP
29
Uni
t Loa
d D
escr
iptio
n En
ergy
With
app
lied
dum
my
load
fP
,0,1
1ki
ii
Ci
ii
FFL
AE
01
1
1ppy 0
i
fk
nF
ii
rr
ir
ki
iC
if
fk
iC
i
CdF
P
FC P
P F P
1ii
i
Rea
l loa
dIm
agin
ary
load
01
MMMdz
EI
TT
30
1
Ass
ume
unit
load
inst
ead
of
Ci
if
f
PP
01
TTTdz
GJ
Exam
ple
Uni
t Loa
d #1
Ener
gy
Det
erm
ine
disp
lace
men
t at D
0
10
1x
MM
TT
dsds
EI
GJ
31
Exam
ple
Uni
t Loa
d #1
Ener
gy
2
4
02
l
xwlx
wl
dxEI
EI
4 4
111
()
242
11
()
62
y z
wl
EI
GJ
wl
EI
GJ
32
9
Flex
ibili
ty M
etho
d D
escr
iptio
n En
ergy
R
emov
e a
redu
ndan
t mem
ber t
o fo
rmul
ate
a S
D p
robl
em
Sol
ve fo
r dis
plac
emen
ts o
f the
SD
pro
blem
D
eter
min
e re
dund
ant l
oad
that
neg
ate
the
sam
e di
spla
cem
ents
33
Exer
cise
Flex
ibili
ty M
etho
d #1
Ener
gy
0,
1,,
1,
11
jjj
nn
aj
jj
BD
jj
FFL
FFL
AE
AE
34
1,2
1
0
jn
jBD
j
BD
BD
BD
FL
aAE
Xa
Exer
cise
Flex
ibili
ty M
etho
d #2
Ener
gy
2.
714.
82,
BD
BD
PL
La
AE
AE
,
From
0
0.56
Ans
BD
BD
BD
BD
BD
BD
AE
AE
Xa
XP
35
Pote
ntia
l Ene
rgy
Tota
l Pot
entia
l Ene
rgy
Ener
gy
To
tal p
oten
tial e
nerg
y TP
E is
the
sum
of i
ts s
train
(int
erna
l) en
ergy
Uan
d th
e po
tent
ial e
nerg
y V
of th
e ap
plie
d ex
tern
al lo
ads
Ze
ropo
tent
iale
nerg
yat
the
unlo
aded
stat
e
1
1
0
()
TPE
r
nn
rr
rr
y
n
VV
P
UV
Pdy
Py
Ze
ro p
oten
tial e
nerg
y at
the
unlo
aded
sta
te
36
1
TPE
()
r
n
rr
UP
UV
10
Pote
ntia
l Ene
rgy
Stab
ility
Ener
gy
(
)0
UV
37
Exer
cise
TPE
#1En
ergy
Ass
ume
sin
0 at
0
and
Bz
vv
Lv
zz
L
38
22
2
24
24
24
3
and
/0
at
/2
2
sin
24
B
BB
vv
dvdx
zL
Mdv
Udz
EI
EI
dzv
vEI
EI
zU
dzL
LL
Exer
cise
TPE
#2En
ergy
24
34B
BvEI
TPE
UV
Wv
L
39
3
4
3
33
4
4(
)0
4
2 a
s co
mpa
red
to e
xact
sol
utio
n 48
B
BB
B
B
LvEI
UV
vv
L
WL
WL
vEI
EI
Prin
cipl
e of
Sup
erpo
sitio
n D
escr
iptio
nEn
ergy
If th
e bo
dy is
line
arly
ela
stic
, the
ff
tf
bff
ief
fect
of a
num
ber o
f for
ces
is
the
sum
of t
he e
ffect
s of
the
forc
es a
pplie
d se
para
tely
.
40
11
Rec
ipro
cal T
heor
em D
escr
iptio
nEn
ergy
1To
tal d
efle
ctio
n at
poi
nt 1
in th
e di
rect
ion
of
from
all
load
sin
fluen
ce o
r fle
xibi
lity
coef
ficie
ntij
Pa
111
112
21
221
122
22
11
22
... ... ...
j
nn
nn
nn
nnn
n
aP
aP
aP
aP
aP
aP
aP
aP
aP
41
111
121
1
221
222
2
12
n n
nn
nnn
n
aa
aP
aa
aP
aa
aP
ij
jia
a
Exer
cise
Rec
ipro
cal T
heor
em #
1En
ergy
The
800
mm
-long
bea
m is
pro
pped
at
500
mm
, giv
ing
0m
mat
0m
mv
x
0 m
m
at
0 m
m0.
3 m
m
at
100
mm
1.4
mm
at
20
0 m
m2.
5 m
m
at
300
mm
1.9
mm
at
40
0 m
m0
mm
at
50
0 m
m
vx
vx
vx
vx
vx
vx
42
2.
3 m
m
at
600
mm
4.8
mm
at
v
xv
x
B
700
mm
10.6
mm
at
80
0 m
m
Det
erm
ine
whe
n th
e ap
plie
d lo
ads
chan
ge.
vx
Exer
cise
Rec
ipro
cal T
heor
em #
2En
ergy
du
eto
40N
at1
4m
mv
C
d
ue to
30
due
to 4
0 N
at
1.4
mm
due
to 4
0 N
at
1.4
mm
due
to 3
0 N
at
1.4
(3/4
)1.
05 m
m
due
to 1
0 N
at
2.4
(1/
N a
t
4)
D
C C CC
vC
vD
vD
vE
vD
43
,
( d
ue to
10
N a
t )
0.6
mm
1.05
0.6
1.65
C
Ctot
C
alvE
v
1
mm
1.65
tan
300
B