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Chapter 9:
Deflections Using Energy Methods
CIVL3310 STRUCTURAL ANALYSIS
Professor CC Chang
Work and Energy Principles
• Work done by external forces
F
F
ΔF
W   FΔ F dΔ F
F
F
0
F
W
ΔF
1
F  ΔF
2
W  F  ΔF
ΔF
Work and Energy Principles
• Where does the work go?
1
W  F  ΔF
2
Saved in the beam in terms of
“Strain energy”
Member
force
fi
i-th component
fi
F
Work - Energy
1
1
F  Δ F   fi  d i
2
2
Member
deformation
1
U i  fi  d i
2
F
WU
di
Total strain energy
1
U   fi  d i
2
Virtual Work Principle
• Under equilibrium, perturb the
 δfi  δd
structure
i
fi
Work - Energy
di
WU
1
1
F  Δ F   fi  d i
2
2
F
δW  δF  ΔF
F
 δF
δF
δΔF
F
δW  δU
δW  F  δΔF
1
W  F  ΔF
2
ΔF
 δΔF
Perturb DF by d DF
FδD F  fi δdi
Perturb F by dF
δFD F  δfi di
Virtual Work Principle
F
δW  δU
F
 δΔF
 δF
(Complementary) virtual work principle
Perturb F by dF
δFD F  δfi di
Virtual work principle
Perturb DF by d DF
FδD F  fi δdi
Virtual Work Principle
(Complementary) virtual work principle
Perturb F by dF
 δfi
fi di
F
F
δW  δU
 δF
δFD F  δfi di
The complementary virtual work done by an external
virtualforce
force
system
external virtual
system
under the actual
actualdeformation
deformationofof
a structure
is equal
a structure
is equal
to to
the complementary strain energy done by the virtual
virtualstresses
stresses
under the actual
actualstrains
strains
Virtual Work Principle
δFD F  δfi di
 δfi
fi di
Superposition
F
Actual system
F
 δF
Virtual system
Virtual Work Principle
Looking for DF
f i Li
1 D F   δfi 
Ei Ai
δFD F  δfi di
δW  δU
1
Virtual system
Actual system
fi
F
F
di
fiLi
EiAi
 δfi
 δF
1
Virtual Work Principle
δW  δU
dfi
fi
F
f i Li
1 D F   δfi 
Ei Ai
F
δF  1
Virtual Work Principle
δW  δU
Temperature effect
dfi
DT
δF  1
F
di  α ΔT Li
1 D F   δfi  di
Virtual Work Principle
δW  δU
Misfit effect
dfi
DL
δF  1
F
1 D F   δfi  di
Virtual Work Principle
• For Beams
M ≠ 0, ≠0, k ≠ 0
dM(x)
M(x)
dF
M=0, ≠0, k=0
DFF
x
d
1
U  M(x)  d
2
Perturbation
dU  dM(x) d
Integrate for the whole beam
M(x)
d 
dx
EI
U
1L
1 L M2
20
2 0 EI
 M(x)  d 
δW  δU

M(x)
L
L
M
dU   dM  d   dM  dx
EI
0
0
 dx
L
M
dF  Δ F   dM  dx
EI
0
Virtual Work Principle
δW  δU
w
P
DF
dF
M(x)
Deformation due to
actual loads
A
dM(x)
dMA
Moment due to
actual loads
L
M
dF  Δ F   dM  dx
EI
0
Virtual force corresponds
to actual deformation
Virtual moment induced by
The virtual force
L
M
dM A  A   dM  dx
EI
0
Virtual Work Principle
• For Frames
δW  δU
w
DF
P
dF
fL
M
dF  Δ F 
df  


 dM  dx
EI component
EI
component
Negligible
Castigliano’s Principle
• Work-Energy
 δfi
fi d i
W Fi   U Fi 
W Fi  dFi   U Fi  dFi 
U
W
dFi
dFi  UFi  
W Fi  
Fi F
Fi F
i
i
U
W

Fi F Fi F
i
i
Fi
Fi
 dFi
Fi
W Fi
  Δ Fi
Fi
k
U
D Fi 
Fi F
i
k
1
1
W  Fi  Δ Fi
2
F F 
 i  i 
2 k
Δ
Fi
1
U   fi  d i
n 2
f i  f i Li 
  

n 2  EA 
fi Fi 
Castigliano’s Principle
• For Trusses
U
D Fi 
Fi F
U
i
f j2L j
2E jA j
 f j  f jL j

D Fi   
 Fi  E jA j
• For Beams
U
D Fi 
Fi F
M2
U
dx
2EI
i
 M  M
 dx
D Fi   
 Fi  EI
• For Frames
 f j  f jL j
 M  M

 dx
D Fi   
  
 Fi  EI
 Fi  E jA j
Betti’s Law of Reciprocal Deflections
L
M
dF  Δ F   dM  dx
EI
0
δW  δU
Location 1
Location 2
P
Q
MP
Q  DP2   MQ 
dx
EI
D P2
P
Virtual work principle
Q
P  D Q1   M P 
DQ1
MQ
EI
dx
P  DQ1  Q  D P2
If P  Q then
D P2  DQ1
Betti’s Law
Betti’s Law of Reciprocal Deflections
P
DA
P
DB
DA  DB
Betti’s Law of Reciprocal Deflections
P
A
M
DB
P  DB  
MPMM
dx M  A
EI
P  D B  M  A
If P  M (in magnitude) then D B  A (in magnitude)
Betti’s Law and Flexibility Coefficients
Location j
Location i
1
D ji  f ji
1
Flexibility coefficient
Deflection at j due to a unit load at i
Dij  fij
Deflection at i due to a unit load at j
fij  f ji
9. Deflections Using Energy Methods
•
•
•
•
Work-energy principle
Virtual work principle
Castigliano’s principle
Betti’s law

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