03 Slopedeflection
DISPLACEMENT METHOD OF ANALYSIS:
SLOPE DEFLECTION EQUATIONS
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General Case
Stiffness Coefficients
Stiffness Coefficients Derivation
Fixed-End Moments
Pin-Supported End Span
Typical Problems
Analysis of Beams
Analysis of Frames: No Sidesway
Analysis of Frames: Sidesway
1
Slope – Deflection Equations
P
i
w
j
k
Cj
settlement = ∆j
Mij
P
i
w
j
Mji
θi ψ θj
2
Degrees of Freedom
M
θΑ
A
B
1 DOF: θΑ
C
2 DOF: θΑ , θΒ
L
θΑ
A
P
B
θΒ
3
Stiffness
kAA
kBA
1
B
A
L
k AA =
4 EI
L
k BA =
2 EI
L
4
kBB
kAB
A
B
1
L
k BB =
4 EI
L
k AB =
2 EI
L
5
Fixed-End Forces
Fixed-End Moments: Loads
P
L/2
PL
8
L/2
PL
8
L
P
2
P
2
w wL2 12 wL 2
L
wL2
12
wL
2
6
General Case
P
i
w
j
k
Cj
settlement = ∆j
Mij
P
i
w
j
Mji
θi ψ θj
7
Mij
P
i
j
w
Mji
θi
L
θj
ψ
4 EI
2 EI θi + θj = M ij L
L
Mji =
2 EI
4 EI θi + θj L
L
θj
θi
(MFij)∆
+
(MFji)∆
settlement = ∆j
+
P
(MFij)Load
M ij = (
settlement = ∆j
w
(MFji)Load
4 EI
2 EI
2 EI
4 EI
)θ i + (
)θ j + ( M F ij ) ∆ + ( M F ij ) Load , M ji = (
)θ i + (
)θ j + ( M F ji ) ∆ + ( M F ji ) Load 8
L
L
L
L
Equilibrium Equations i P
w
j
k
Cj
Cj M
Mji
Mji
jk
Mjk j + ΣM j = 0 : − M ji − M jk + C j = 0
9
Stiffness Coefficients
Mij
i
j
Mji
L
θj
θi
kii =
4 EI
L
k ji =
2 EI
L
×θ i
k jj =
4 EI
L
×θ j
1
kij =
2 EI
L
+
1
10
Matrix Formulation
M ij = (
4 EI
2 EI
)θ i + (
)θ j + ( M F ij )
L
L
M ji = (
2 EI
4 EI
)θ i + (
)θ j + ( M F ji )
L
L
M ij (4 EI / L) ( 2 EI / L) θ iI M ij F
M =
θ + M F
EI
L
EI
L
(
2
/
)
(
4
/
)
j ji
ji
kii
k ji
[k ] =
kij k jj
Stiffness Matrix
11
P
i
Mij
w
j
Mji
θi
L
[ M ] = [ K ][θ ] + [ FEM ]
θj
ψ
∆j
([ M ] − [ FEM ]) = [ K ][θ ]
[θ ] = [ K ]−1[ M ] − [ FEM ]
Mij
Mji
θj
θi
Fixed-end moment
Stiffness matrix matrix
+
(MFij)∆
(MFji)∆
[D] = [K]-1([Q] - [FEM])
+
(MFij)Load
P
w
(MFji)Load
Displacement matrix Force matrix
12
SLOPE DEFLECTION EQUATIONS
!
!
!
!
!
!
!
!
!
General Case
Stiffness Coefficients
Stiffness Coefficients Derivation
Fixed-End Moments
Pin-Supported End Span
Typical Problems
Analysis of Beams
Analysis of Frames: No Sidesway
Analysis of Frames: Sidesway
1
Slope – Deflection Equations
P
i
w
j
k
Cj
settlement = ∆j
Mij
P
i
w
j
Mji
θi ψ θj
2
Degrees of Freedom
M
θΑ
A
B
1 DOF: θΑ
C
2 DOF: θΑ , θΒ
L
θΑ
A
P
B
θΒ
3
Stiffness
kAA
kBA
1
B
A
L
k AA =
4 EI
L
k BA =
2 EI
L
4
kBB
kAB
A
B
1
L
k BB =
4 EI
L
k AB =
2 EI
L
5
Fixed-End Forces
Fixed-End Moments: Loads
P
L/2
PL
8
L/2
PL
8
L
P
2
P
2
w wL2 12 wL 2
L
wL2
12
wL
2
6
General Case
P
i
w
j
k
Cj
settlement = ∆j
Mij
P
i
w
j
Mji
θi ψ θj
7
Mij
P
i
j
w
Mji
θi
L
θj
ψ
4 EI
2 EI θi + θj = M ij L
L
Mji =
2 EI
4 EI θi + θj L
L
θj
θi
(MFij)∆
+
(MFji)∆
settlement = ∆j
+
P
(MFij)Load
M ij = (
settlement = ∆j
w
(MFji)Load
4 EI
2 EI
2 EI
4 EI
)θ i + (
)θ j + ( M F ij ) ∆ + ( M F ij ) Load , M ji = (
)θ i + (
)θ j + ( M F ji ) ∆ + ( M F ji ) Load 8
L
L
L
L
Equilibrium Equations i P
w
j
k
Cj
Cj M
Mji
Mji
jk
Mjk j + ΣM j = 0 : − M ji − M jk + C j = 0
9
Stiffness Coefficients
Mij
i
j
Mji
L
θj
θi
kii =
4 EI
L
k ji =
2 EI
L
×θ i
k jj =
4 EI
L
×θ j
1
kij =
2 EI
L
+
1
10
Matrix Formulation
M ij = (
4 EI
2 EI
)θ i + (
)θ j + ( M F ij )
L
L
M ji = (
2 EI
4 EI
)θ i + (
)θ j + ( M F ji )
L
L
M ij (4 EI / L) ( 2 EI / L) θ iI M ij F
M =
θ + M F
EI
L
EI
L
(
2
/
)
(
4
/
)
j ji
ji
kii
k ji
[k ] =
kij k jj
Stiffness Matrix
11
P
i
Mij
w
j
Mji
θi
L
[ M ] = [ K ][θ ] + [ FEM ]
θj
ψ
∆j
([ M ] − [ FEM ]) = [ K ][θ ]
[θ ] = [ K ]−1[ M ] − [ FEM ]
Mij
Mji
θj
θi
Fixed-end moment
Stiffness matrix matrix
+
(MFij)∆
(MFji)∆
[D] = [K]-1([Q] - [FEM])
+
(MFij)Load
P
w
(MFji)Load
Displacement matrix Force matrix
12