1
!
Development: The Slope-Deflection
Equations
!
Stiffness Matrix
!
General Procedures
!
Internal Hinges
!
Temperature Effects
!
Force & Displacement Transformation
!
Skew Roller Support
BEAM ANALYSIS USING THE STIFFNESS
METHOD
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06 matrix beam hinge, Study notes of Civil Engineering
matrix method, how to account for hinge
Typology: Study notes
2015/2016
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!
Development: The Slope-DeflectionEquations
!
Stiffness Matrix
!
General Procedures
!
Internal Hinges
!
Temperature Effects
!
Force & Displacement Transformation
!
Skew Roller Support
BEAM ANALYSIS USING THE STIFFNESSMETHOD
Slope ñ Deflection Equations
settlement =
j
P
i
j
k
w
C
j
M
ij
M
ji
w
P
j
i
i
j
L
A
B
ï
Stiffness Definition
k
BA
k
AA
EI L
k
AA
EI L
k
BA
L
A
B
k
BB
k
AB
EI L
k
BB
EI L
k
AB
ï
General Case
settlement =
j
P
i
j
k
w
C
j
M
ij
M
ji
w
P
j
i
i
j
w
P
settlement =
j
(M
F
ij
∆
(M
F
ji
∆
(M
F
ij
Load
(M
F
ji
Load
M
ij
M
ji
i
j
i
j
w
P
M
ij
M
ji
settlement =
j
j
i
j
i^
EI L
EI L
j
i^
EI L
EI L
Load
ij F
ij F
j
i
ij
M
M
EI L
EI L
M
∆
Load
ji F
ji F
j
i
ji
M
M
EI L
EI L
M
∆
L
i
j
M
ij
M
ji
i
j
EI L
k
ii
EI L
k
ji
EI L
k
ij
EI L
k
jj
i
×
j
×
ï
Stiffness Coefficients
L
[ ]
jj
ji
ij
ii
k
k
k
k
k
Stiffness Matrix
ij F
j
i
ij
M
EI L
EI L
M
ji F
j
i
ji
M
EI L
EI L
M
F ji
F ij
j iI
ij ji
M M
L
EI
L
EI
L
EI
L
EI
M M
θ^ θ
) / 2 ( ) / 4 ( ï
Matrix Formulation
L
Real beam Conjugate beam
ï
Stiffness Coefficients Derivation
M
j
M
i
L
i
L
M
M
j
i^
M^ EI
j
M EI
i
ι
EI
L
M
j
EI
L
M
i
2
j
i
j
i
i
M
M
L
EI
L
M
L
EI
L
M
M
E
L I
M
EI
L
M
F
j
i
i
y
i
j
i
i
EI L
M
EI L
M
and
From
θ^ θ
=^ =
L
M
M
j
i^
i
j
ï
Derivation of Fixed-End Moment
Real beam
2
PL
M
EI
PL
EI
ML
EI
ML
F
y
= = + − − = Σ ↑ + P
M
M
M EI
Conjugate beam
A
M EI
B
L
P
A
B
M EI
EI
ML 2
M EI
EI
ML 2
E
I
PL
2
EI
PL
E
I
PL
2
Point load
Uniform load
L
w
A
B
w
M
M
Real beam
Conjugate beam
A
M EI
M EI
B
2
3
wL
M
EI
wL
EI
ML
EI
ML
F
y
E
I
wL 8
2
EI
wL 24
3
EI
wL 24
3
M EI
EI
ML 2
M EI
EI
ML 2
17
Settlements
M
M
L
M
j
M
i^
M
j
L
M
M
j
i^
L
M
M
j
i^
Real beam
2
EI^ L
M
Conjugate beam
M EI
A
B
M EI
L
EI
ML
L
EI
ML
M
B
M EI
EI
ML 2
M EI
EI
ML 2
M
BA
M
BC
A
C
B
P
1
P
2
L
1
L
2
w C
B
B
C
B
1
1
1
1
L
P
EI L
EI L
M
B
A
BA
2 2 2 2 2 2
wL
L
P
EI L
EI L
M
C
B
BC
B
BC
BA
B
B
for
Solve
M
M
C
M
A
C
B
P
1
P
2
L
1
L
2
w C
B
Substitute
B
in
M
AB
M
BA
, M
BC
, M
CB
M
AB
M
BA
M
BC
M
CB
1
1
1
1
L
P
EI L
EI L
M
B
A
AB
1
1
1
1
L
P
EI L
EI L
M
B
A
BA
2 2 2 2 2 2
wL
L
P
L
EI
EI L
M
C
B
BC
2 2 2 2 2 2
wL
L
P
L
EI
EI L
M
C
B
CB