Page 1
Structural Analysis Quick Reference & Lecture Notes
Planar Trusses:
Loads: applied at the joints only
Internal Forces: axial tension or compressions
Method of Joints: 2 Equilibrium Equations per joint
Method of Sections: 3 Equilibrium Equations per Section
Sign Convention: Assume that unknown internal forces are tensile
(pulling away from the member)
Beams:
Loads: Concentrated forces, couples, distributed loads
Internal Forces
: shear forces and bending moments
Equilibrium: 3 equations for whole beam or section of beam (e.g. 2 force
Eqs. & 1 Moment Eq. or 1 Force Eq. and 2 Moment Eqs. etc…)
Sign Convention: shear forces creating clockwise couple are positive
bending moments creating compression at top of member are positive
Bending moment drawn on compression side of member.
Planar Frames
Loads: Concentrated forces, couples, distributed loads
Internal Forces: axial forces, shear forces and bending moments
Equilibrium:
3 equations for whole frame or section of frame (e.g. 2 force
Eqs. & 1 Moment Eq. or 1 Force Eq. and 2 Moment Eqs. etc…)
Sign Convention: same as for beam with frame outside treated same as
top of beam (positive side)
Support Reactions & Displacements:
Fixed Support: 3 reactions & 0 displacement
Simple Support: 2 reactions & 1 displacement
Roller Support: 1 reaction & 2 displacements
No Support: 0 reaction & 3 displacements
∑
∑
== 00
yx
FF
Prof. M. Maalej
Page 2
Principle of Conservation of Energy
U
e
= U
i
Unit Load Method (ULM)
Moment Equation for ULM
P
x
= ?
Example:
P
2
P
1
d
1
d
Structural Analysis Quick Reference & Lecture Notes
Prof. M. Maalej
Page 3
Force Method
Beams & Frames
Trusses
Slope Deflection Method
1. Determine No. of DOF (Unknown Displ.)
2. Compute Fixed-End Moments (FEM)
3. Write Slope-Deflection Equations
4. Write Equilibrium Equations at the
joints and Shear Equation when
applicable (e.g. M
BA
+ M
BD
= 0)
5. Solve for the unknow displacements (e.g.
B
,
D
, )
6. Compute the End Moments: M
AB
, M
BA
, etc…
7. Compute Shear Forces: V
AB
, V
BA
, etc…
8. Compute support reactions when needed
9. Draw Shear & Moments Diagrams when needed
Modified Slope-Deflection Method
( )
( )
BABABA
ABBAAB
FEM
L
EI
M
FEM
L
EI
M
+−+=
+−+=
32
2
32
2
Chord Rotation
=
AB
=
BA
= -
( )
0
2
3
=
−+−=
AB
AB
BABBA
M
FEM
FEM
L
EI
M
AB
B
A
FEM
EI
L
42
3
2
−+−=
Structural Analysis Quick Reference & Lecture Notes
Prof. M. Maalej
Page 4
Stiffness Method
Structural Analysis Quick Reference & Lecture Notes
Prof. M. Maalej
Page 5
Structural Analysis Quick Reference & Lecture Notes
Refer to member stiffness templates and worksheets for k
i
expressions
Prof. M. Maalej
Beam Member Stiffness Matrix
−
−−−
−
−
=
3322
3322
22
22
121266
121266
6642
6624
L
EI
L
EI
L
EI
L
EI
L
EI
L
EI
L
EI
L
EI
L
EI
L
EI
L
EI
L
EI
L
EI
L
EI
L
EI
L
EI
k
m
1
2
3
4
1
2
3
4
p
1
m
, x
1
m
p
2
m
, x
2
m
p
3
m
, x
3
m
p
4
m
, x
4
m
EI, L
A
B
Member “m”
p
1
m
, x
1
m
p
2
m
, x
2
m
p
3
m
, x
3
m
p
4
m
, x
4
m
EI, L
A
B
p
1
m
, x
1
m
p
2
m
, x
2
m
p
3
m
, x
3
m
p
4
m
, x
4
m
p
1
m
, x
1
m
p
2
m
, x
2
m
p
3
m
, x
3
m
p
4
m
, x
4
m
EI, L
A
B
Member “m”
k
1
for member 1
k
2
for member 2
k
3
for member 3
k
4
for member 4
EI = L =
EI = L =
EI = L =
EI = L =
Prof. M. Maalej
Page 6
Truss Member Stiffness Matrix
−
−
−−
−
−
−−
θθθθθθ
θθθθθθ
θθθθθθ
θθθθθθ
22
22
22
22
sincossinsincossin
cossincoscossincos
sincossinsincossin
cossincoscossincos
L
EA
L
EA
L
EA
L
EA
L
EA
L
EA
L
EA
L
EA
L
EA
L
EA
L
EA
L
EA
L
EA
L
EA
L
EA
L
EA
k
m
for member m
1
2
3
4
1
2
3
4
k
2
for member 2
k
3
for member 3
k
4
for member 4
EA = L = EA/L =
cos θ = sin θ = cos θ sin θ =
cos
2
θ = sin
2
θ =
EA = L = EA/L =
cos θ = sin θ = cos θ sin θ =
cos
2
θ = sin
2
θ =
EA = L = EA/L =
cos θ = sin θ = cos θ sin θ =
cos
2
θ = sin
2
θ =
k
1
for member 1
EA = L = EA/L =
cos θ = sin θ = cos θ sin θ =
cos
2
θ = sin
2
θ =
Prof. M. Maalej
Page 7
SC ==
θθ
sincos
Frame Member Stiffness Matrix
k
1
=====
====
====
SCSCSC
LEILEILEILEI
LEALEAEI
22
32
12624
k
2
=====
====
====
SCSCSC
LEILEILEILEI
LEALEAEI
22
32
12624
+
+−−−
−
+−+
−
−−−
−
−
−
−−
−−+
+
−−
−−−
+
−
+
−−
=
θθθθθθθθθθ
θθθθθθθθθθ
θθθθ
θθθθθθθθθθ
θθθθθθθθθθ
θθθθ
22
332
22
332
3
22
323
22
32
2222
22
332
22
332
3
22
323
22
32
2222
sincos
12
cossin
12
cos
6
sincos
12
cossin
12
cos
6
cossin
12
cossin
12
sin
6
cossin
12
cossin
12
sin
6
cos
6
sin
64
cos
6
sin
62
sincos
12
cossin
12
cos
6
sincos
12
cossin
12
cos
6
cossin
12
cossin
12
sin
6
cossin
12
cossin
12
sin
6
cos
6
sin
62
cos
6
sin
64
L
EA
L
EI
L
EA
L
EI
L
EI
L
EA
L
EI
L
EA
L
EI
L
EI
L
EA
L
EI
L
EA
L
EI
L
EI
L
EA
L
EI
L
EA
L
EI
L
EI
L
EI
L
EI
L
EI
L
EI
L
EI
L
EI
L
EA
L
EI
L
EA
L
EI
L
EI
L
EA
L
EI
L
EA
L
EI
L
EI
L
EA
L
EI
L
EA
L
EI
L
EI
L
EA
L
EI
L
EA
L
EI
L
EI
L
EI
L
EI
L
EI
L
EI
L
EI
L
EI
m
k
Prof. M. Maalej
Page 8
