Download - CIVL3310 STRUCTURAL ANALYSIS Professor CC Chang Chapter 9: Deflections Using Energy Methods
CIVL3310 STRUCTURAL ANALYSISProfessor CC Chang
Chapter 9: Deflections Using Energy Methods
Work and Energy Principles• Work done by external forces
FF
F
FΔ
FΔF2
1W
F
FΔ
FΔFW
FΔ
0FF ΔΔFW d
F
Work and Energy Principles• Where does the work go?
Saved in the beam in terms of
“Strain energy”
FF
i-th componentif
if
Member force
Member deformation
id
iii df2
1U
FΔF2
1W
ii df2
1UTotal strain energy
Work - Energy
UW
iiF df2
1ΔF
2
1
Virtual Work Principle• Under equilibrium, perturb the
structure
FF
F
FΔ
FδΔ
δF
FδΔFδW
FΔδFδW
FΔF2
1W
Work - Energy
UW
iiF df2
1ΔF
2
1
Perturb F by dF
Perturb DF by d DFδUδW
iiF dδfδF
iiF δdfδF
FδΔ
δF
ii df iδdiδf
Virtual Work Principle
(Complementary) virtual work principlePerturb F by dF
Virtual work principlePerturb DF by d DF
δUδW
iiF dδfδF
iiF δdfδF
F
F FδΔ
δF
Virtual Work Principle(Complementary) virtual work principlePerturb F by dF
δUδW
iiF dδfδF
FF
δF
iδf
The complementary virtual work done by an external virtual force system under the actual deformation of a structure is equal to the complementary strain energy done by the virtual stresses under the actual strains
The complementary virtual work done by an external virtual force system under the actual deformation of a structure is equal to the complementary strain energy done by the virtual stresses under the actual strains
fi di
Virtual Work Principle
iiF dδfδF
FF
δF
iδffi di
Superposition
Actual system Virtual system
Virtual Work Principle
iiF dδfδF
FF δF
iδffi di
Actual system Virtual system
δUδW
Looking for DF
1
1
fiLi
EiAi
ii
iiiF AE
Lfδf1
Virtual Work Principle
ii
iiiF AE
Lfδf1
δUδW
F
F
1δF fi
d fi
Virtual Work Principle
iiF dδf1
δUδW
F
1δF
d fiDT
Temperature effect
ii LΔTαd
Virtual Work Principle
iiF dδf1
δUδW
F
1δF
d fiDL
Misfit effect
Virtual Work Principle• For Beams
F
dM(x)2
1U
M=0, q≠0, k=0
M ≠ 0, q≠0, k ≠ 0M(x)
xdq
M(x)
dd dM(x)UIntegrate for the whole beam
LL
dd0
2
0x
EI
M
2
1M(x)
2
1U
Perturbation
xEI
M(x)dd
d ddLL
dd00
xEI
MMMU
DF
δUδW ddL
d0
F xEI
MMΔF
dFdM(x)
Virtual Work Principle
δUδW
ddL
d0
F xEI
MMΔF
P
dF
w
Deformation due to actual loads
Moment due to actual loads
DF
M(x) dM(x)
Virtual force corresponds to actual deformation
Virtual moment induced by The virtual force
ddL
d0
AA xEI
MMM
qA
dMA
Virtual Work Principle• For Frames
δUδW
d ddcomponentcomponent
dxEI
MM
EI
fLfΔF F
Negligible
w
DF
Castigliano’s Principle• Work-Energy ii FUFW
Fi
Fi
idF
ii dfiδf
Fi
iFΔ
k
F
2
F
ΔF2
1W
ii
Fi i
1
k
n
n
EA
Lf
2
f
df2
1U
iii
ii ii Ff
i
i
ii
ii
FiF
FiFi
iFi
iiFi
i
iiii
F
U
F
U
F
W
dFF
UFUdF
F
WFW
dFFUdFFW
iFi
iΔ
k
F
F
W
Castigliano’s Principle• For Trusses
• For Beams
• For Frames
jj
jj
i
jF
jj
j2
j
FiF AE
Lf
F
f
A2E
LfU
F
Ui
i
i
dxEI
M
F
Mdx
2EI
MU
F
U
iF
2
FiF i
i
i
dxEI
M
F
M
AE
Lf
F
f
ijj
jj
i
jFi
Betti’s Law of Reciprocal Deflections
P
Q
2P
Location 1 Location 2
1Q
dxEI
MMQ P
QP2
Virtual work principle
dxEI
MMP Q
PQ1
21 PQ QP
Betti’s Law
Q
P
thenQ PIf 12 QP
δUδW ddL
d0
F xEI
MMΔF
Betti’s Law of Reciprocal Deflections
A
P
B
P
BA
Betti’s Law of Reciprocal Deflections
A
P
B
M
AB MP
AMP
B MdxEI
MMP
thenmagnitude) (inM PIf magnitude) (in AB
Betti’s Law and Flexibility Coefficients
jiji f
1
1
jiij ff
Location j Location i
ijij f
Deflection at j due to a unit load at i
Flexibility coefficient
Deflection at i due to a unit load at j
9. Deflections Using Energy Methods
• Work-energy principle• Virtual work principle• Castigliano’s principle• Betti’s law