Formula Sheets: Matrix Method of Analysis | Structural Analysis - Civil Engineering (CE) PDF Download

 Page 1


GA TE CE 2026 F orm ula Sheet: Displacemen t Metho ds and Struc-
tural Analysis
1. Slop e Deflection Metho d
• Slop e deflection equation for a mem b er:
M
ij
=
2EI
L
(
2?
i
+?
j
-
3?
L
)
+M
Fij
where:
– M
ij
: Momen t at end i of mem b er ij (kN·m)
– E : Mo dulus o f elasticit y (N/mm²)
– I : Momen t of inertia (mm?)
– L : Length of mem b er (m)
– ?
i
, ?
j
: Rotations at ends i and j (radians)
– ? : Relativ e displacemen t b et w een ends (mm)
– M
Fij
: Fixed-end momen t at end i (kN·m)
• Fixed-end momen ts (examples):
UDL: M
Fij
=-
wL
2
12
, M
Fji
=
wL
2
12
P oin t load at midspan: M
Fij
=-
PL
8
, M
Fji
=
PL
8
where:
– w : Uniformly distributed l oad (kN/m)
– P : P oin t load (kN)
2. Momen t Distribution Metho d
• Distribution factor:
DF
ij
=
k
ij
?
k
where:
– DF
ij
: Dis tribution factor for mem b er ij
– k
ij
: Stiffness of mem b er ij , k
ij
=
4EI
L
(fixed end) or
3EI
L
(pinned end)
–
?
k : Sum of stiffnesses of all mem b ers at join t
• Carry-o v er factor:
CO = 0.5 ( for fixed ends )
• Distributed momen t:
M
d,ij
= DF
ij
·M
un balanced
where:
– M
d,ij
: Distributed momen t to mem b er ij (kN·m)
– M
un balanced
: Un balanced momen t at join t (kN·m)
• Final momen t:
M
ij
=M
Fij
+
?
M
d,ij
1
Page 2


GA TE CE 2026 F orm ula Sheet: Displacemen t Metho ds and Struc-
tural Analysis
1. Slop e Deflection Metho d
• Slop e deflection equation for a mem b er:
M
ij
=
2EI
L
(
2?
i
+?
j
-
3?
L
)
+M
Fij
where:
– M
ij
: Momen t at end i of mem b er ij (kN·m)
– E : Mo dulus o f elasticit y (N/mm²)
– I : Momen t of inertia (mm?)
– L : Length of mem b er (m)
– ?
i
, ?
j
: Rotations at ends i and j (radians)
– ? : Relativ e displacemen t b et w een ends (mm)
– M
Fij
: Fixed-end momen t at end i (kN·m)
• Fixed-end momen ts (examples):
UDL: M
Fij
=-
wL
2
12
, M
Fji
=
wL
2
12
P oin t load at midspan: M
Fij
=-
PL
8
, M
Fji
=
PL
8
where:
– w : Uniformly distributed l oad (kN/m)
– P : P oin t load (kN)
2. Momen t Distribution Metho d
• Distribution factor:
DF
ij
=
k
ij
?
k
where:
– DF
ij
: Dis tribution factor for mem b er ij
– k
ij
: Stiffness of mem b er ij , k
ij
=
4EI
L
(fixed end) or
3EI
L
(pinned end)
–
?
k : Sum of stiffnesses of all mem b ers at join t
• Carry-o v er factor:
CO = 0.5 ( for fixed ends )
• Distributed momen t:
M
d,ij
= DF
ij
·M
un balanced
where:
– M
d,ij
: Distributed momen t to mem b er ij (kN·m)
– M
un balanced
: Un balanced momen t at join t (kN·m)
• Final momen t:
M
ij
=M
Fij
+
?
M
d,ij
1
3. Influence Lines
• Influence line ordinate for reaction (simply supp orted b eam):
IL
R
(x) = 1-
x
L
where:
– IL
R
(x) : Influ ence line ordinate for reaction at supp ort
– x : Distance from supp ort (m)
– L : Span length (m)
• Influence line ordinate for shear at section:
IL
V
(x) =
{
1-
x
L
if x<x
s
-
x
L
if x=x
s
where:
– x
s
: P osition of section (m)
• Influence line ordinate for momen t at section:
IL
M
(x) =
x(L-x
s
)
L
• Maxim um resp onse due to mo ving load:
R
max
=
?
P
i
·IL
i
where:
– R
max
: Maxim um reaction, she ar,op olitical or momen t
– P
i
: Magnitude of mo ving load i (kN)
– IL
i
: Influence line ordinate at load p osition
4. Stiff ness Metho d
• Mem b er stiffness matrix (b eam elemen t, 2D):
[k] =
EI
L
3
?
?
?
?
12 6L -12 6L
6L 4L
2
-6L 2L
2
-12 -6L 12 -6L
6L 2L
2
-6L 4L
2
?
?
?
?
where:
– [k] : Mem b er stiffness matrix
– Ro ws/columns: V ertical displacemen t, rotation at end 1, v ertical displacemen t, rotation at
end 2
• Global stiffness matrix:
[K] =
?
[k
transformed
]
• Equilibrium equation:
[K]{?} ={F}
where:
– {?} : Displacemen t v ector
– {F} : F orce v ector
• Mem b er forces:
{f} = [k]{d}
where:
– {f} : Mem b er forces (shear, momen t)
– {d} : Mem b er displac emen ts
2
Page 3


GA TE CE 2026 F orm ula Sheet: Displacemen t Metho ds and Struc-
tural Analysis
1. Slop e Deflection Metho d
• Slop e deflection equation for a mem b er:
M
ij
=
2EI
L
(
2?
i
+?
j
-
3?
L
)
+M
Fij
where:
– M
ij
: Momen t at end i of mem b er ij (kN·m)
– E : Mo dulus o f elasticit y (N/mm²)
– I : Momen t of inertia (mm?)
– L : Length of mem b er (m)
– ?
i
, ?
j
: Rotations at ends i and j (radians)
– ? : Relativ e displacemen t b et w een ends (mm)
– M
Fij
: Fixed-end momen t at end i (kN·m)
• Fixed-end momen ts (examples):
UDL: M
Fij
=-
wL
2
12
, M
Fji
=
wL
2
12
P oin t load at midspan: M
Fij
=-
PL
8
, M
Fji
=
PL
8
where:
– w : Uniformly distributed l oad (kN/m)
– P : P oin t load (kN)
2. Momen t Distribution Metho d
• Distribution factor:
DF
ij
=
k
ij
?
k
where:
– DF
ij
: Dis tribution factor for mem b er ij
– k
ij
: Stiffness of mem b er ij , k
ij
=
4EI
L
(fixed end) or
3EI
L
(pinned end)
–
?
k : Sum of stiffnesses of all mem b ers at join t
• Carry-o v er factor:
CO = 0.5 ( for fixed ends )
• Distributed momen t:
M
d,ij
= DF
ij
·M
un balanced
where:
– M
d,ij
: Distributed momen t to mem b er ij (kN·m)
– M
un balanced
: Un balanced momen t at join t (kN·m)
• Final momen t:
M
ij
=M
Fij
+
?
M
d,ij
1
3. Influence Lines
• Influence line ordinate for reaction (simply supp orted b eam):
IL
R
(x) = 1-
x
L
where:
– IL
R
(x) : Influ ence line ordinate for reaction at supp ort
– x : Distance from supp ort (m)
– L : Span length (m)
• Influence line ordinate for shear at section:
IL
V
(x) =
{
1-
x
L
if x<x
s
-
x
L
if x=x
s
where:
– x
s
: P osition of section (m)
• Influence line ordinate for momen t at section:
IL
M
(x) =
x(L-x
s
)
L
• Maxim um resp onse due to mo ving load:
R
max
=
?
P
i
·IL
i
where:
– R
max
: Maxim um reaction, she ar,op olitical or momen t
– P
i
: Magnitude of mo ving load i (kN)
– IL
i
: Influence line ordinate at load p osition
4. Stiff ness Metho d
• Mem b er stiffness matrix (b eam elemen t, 2D):
[k] =
EI
L
3
?
?
?
?
12 6L -12 6L
6L 4L
2
-6L 2L
2
-12 -6L 12 -6L
6L 2L
2
-6L 4L
2
?
?
?
?
where:
– [k] : Mem b er stiffness matrix
– Ro ws/columns: V ertical displacemen t, rotation at end 1, v ertical displacemen t, rotation at
end 2
• Global stiffness matrix:
[K] =
?
[k
transformed
]
• Equilibrium equation:
[K]{?} ={F}
where:
– {?} : Displacemen t v ector
– {F} : F orce v ector
• Mem b er forces:
{f} = [k]{d}
where:
– {f} : Mem b er forces (shear, momen t)
– {d} : Mem b er displac emen ts
2
5. Flexibilit y Metho d
• Flexibilit y matrix for a mem b er:
d
ij
=
?
M
i
M
j
EI
dx
where:
– d
ij
: Flexibilit y co e?icien t (deflection due to unit forc e)
– M
i
, M
j
: Momen t functions due to unit loads
• Compatibilit y equation:
[F]{X}+{?
0
} ={0}
where:
– [F] : Flexibilit y matrix
– {X} : Redundan t forces
– {?
0
} : Displacemen ts due to applied loads
• Mem b er forces:
M =M
0
+
?
X
i
M
i
where:
– M
0
: Momen t due to applied loads
– M
i
: Momen t due to redundan t force X
i
3