Formula Sheets: Free Body Diagrams & Equilibrium Equations

 Page 1


GA TE CE 2026 F orm ula Sheet: Engineering Mec hanics
1. System of F orces
• Resultan t force (v ector sum):
?
R =
?
?
F
i
=
v
(
?
F
ix
)
2
+
(
?
F
iy
)
2
+
(
?
F
iz
)
2
where:
–
?
R : Resultan t force (N)
–
?
F
i
: Individual force v ectors (N)
– F
ix
, F
iy
, F
iz
: Comp onen ts of force in x, y , z directions
• Direction of resultan t force:
?
x
= cos
-1
(?
F
ix
R
)
, ?
y
= cos
-1
(?
F
iy
R
)
, ?
z
= cos
-1
(?
F
iz
R
)
where:
– ?
x
, ?
y
, ?
z
: Direction cosines (angles with resp ect to axes)
• Momen t of a force ab out a p oin t:
?
M =? r×
?
F
where:
–
?
M : Momen t v ector (N·m)
– ? r : P osition v ector from p oin t to line of a ction of force (m)
–
?
F : F orce v ector (N)
• Scalar momen t ab out an axis:
M =F ·d
where:
– d : P erp endicular distance from axis to l ine of action of force (m)
2. F ree-Bo dy Diagrams and Equilibrium Equations
• Equilibrium conditions (2D):
?
F
x
= 0,
?
F
y
= 0,
?
M
z
= 0
where:
–
?
F
x
,
?
F
y
: Sum of forces in x and y directions (N)
–
?
M
z
: Sum of momen ts ab out z-axis (N·m)
• Equilibrium conditions (3D):
?
F
x
= 0,
?
F
y
= 0,
?
F
z
= 0
?
M
x
= 0,
?
M
y
= 0,
?
M
z
= 0
• Reaction forces at supp orts:
R =
v
R
2
x
+R
2
y
( for 2D)
where:
– R
x
, R
y
: Reaction force comp onen ts (N)
1
Page 2


GA TE CE 2026 F orm ula Sheet: Engineering Mec hanics
1. System of F orces
• Resultan t force (v ector sum):
?
R =
?
?
F
i
=
v
(
?
F
ix
)
2
+
(
?
F
iy
)
2
+
(
?
F
iz
)
2
where:
–
?
R : Resultan t force (N)
–
?
F
i
: Individual force v ectors (N)
– F
ix
, F
iy
, F
iz
: Comp onen ts of force in x, y , z directions
• Direction of resultan t force:
?
x
= cos
-1
(?
F
ix
R
)
, ?
y
= cos
-1
(?
F
iy
R
)
, ?
z
= cos
-1
(?
F
iz
R
)
where:
– ?
x
, ?
y
, ?
z
: Direction cosines (angles with resp ect to axes)
• Momen t of a force ab out a p oin t:
?
M =? r×
?
F
where:
–
?
M : Momen t v ector (N·m)
– ? r : P osition v ector from p oin t to line of a ction of force (m)
–
?
F : F orce v ector (N)
• Scalar momen t ab out an axis:
M =F ·d
where:
– d : P erp endicular distance from axis to l ine of action of force (m)
2. F ree-Bo dy Diagrams and Equilibrium Equations
• Equilibrium conditions (2D):
?
F
x
= 0,
?
F
y
= 0,
?
M
z
= 0
where:
–
?
F
x
,
?
F
y
: Sum of forces in x and y directions (N)
–
?
M
z
: Sum of momen ts ab out z-axis (N·m)
• Equilibrium conditions (3D):
?
F
x
= 0,
?
F
y
= 0,
?
F
z
= 0
?
M
x
= 0,
?
M
y
= 0,
?
M
z
= 0
• Reaction forces at supp orts:
R =
v
R
2
x
+R
2
y
( for 2D)
where:
– R
x
, R
y
: Reaction force comp onen ts (N)
1
3. In t ernal F orces in Structures
• Axial force in a mem b er:
N =
?
F ·cos?
where:
4. Shear F orce and Bending Momen t (Common Cases)
– Simply supp orted b eam with UDL:
V
max
=
wL
2
, M
max
=
wL
2
8
where:
5. F riction F orces
* F rictional force:
F
f
=µ·N
where:
2