Statics – Problem

Le Duong Hung Anh, Ph.D.

leduonghunganh@hcmut.edu.vn
F U N D A M E N TA L C O N C E P T S
Review

Parallelogram Law
Two forces add according to the parallelogram law. The
components form the sides of the parallelogram and
the resultant is the diagonal.

Rectangular Components: Two Dimensions

Engineering Mechanics- Statics Le Duong Hung Anh

F U N D A M E N TA L C O N C E P T S
Problem
Determine the magnitude of the resultant force of the following systems.

Engineering Mechanics- Statics Le Duong Hung Anh

F U N D A M E N TA L C O N C E P T S
Problem
Determine the magnitude of the resultant force of the following systems.

Engineering Mechanics- Statics Le Duong Hung Anh

F U N D A M E N TA L C O N C E P T S
Review

Cartesian Vectors
F A force can be resolved into its Cartesian components along the x, y, z
u=
F
axes so that 𝐹Ԧ = 𝐹𝑥 𝑖Ԧ + 𝐹𝑦 𝑗Ԧ + 𝐹𝑧 𝑘

The magnitude of F is determined from the positive square root of

the sum of the squares of its components.

F = F = Fx2 + Fy2 + Fx2

F Fx F F
u= = i+ y j+ z k
F F F F
u = cos  i + cos  j + cos  k

Engineering Mechanics- Statics Le Duong Hung Anh

F U N D A M E N TA L C O N C E P T S
Problem
Determine the magnitude and coordinate direction angles of the resultant force.

Engineering Mechanics- Statics Le Duong Hung Anh

F U N D A M E N TA L C O N C E P T S
Problem
Determine the magnitude and coordinate direction angles of the resultant force.

Engineering Mechanics- Statics Le Duong Hung Anh

EQUILIBRIUM OF A RIGID BODY
Determine the resultant force of the given force systems.

R = 16.8 ( kN ) i + 3.49 ( kN ) j

The bolt is subjected to the force F, which has components acting along the x,
y, z axes as shown. If the magnitude of F is 80 N, and 𝛼 = 60° and γ = 45°,
determine the magnitudes of its components.

F = 40 ( N ) i + 40 ( N ) j + 56.6 ( N ) k

Engineering Mechanics- Statics Le Duong Hung Anh

EQUILIBRIUM OF A RIGID BODY
Condition for the Equilibrium of a Particle

For equilibrium of a 2-dimensional concurrent coplanar force system, these

forces must sum to produce a zero force resultant

F = 0 ⟹
F
x =0
⟹ Two unknowns
F i + F
x y j =0 F
y =0

F x =0
F = 0 ⟹ F =0 ⟹ Three unknowns
F i + F j +  Fz k = 0
y

F =0
x y
z

Engineering Mechanics- Statics Le Duong Hung Anh

 FUNDAMENTAL CONCEPTS
Moment of a force about a point

 = 600
( )
mO F = ??
Engineering Mechanics- Statics Le Duong Hung Anh
 F U N DA M E N TA L C O N C E P T S
Moment of a force about a point

( )
mO F = ??

Engineering Mechanics- Statics Le Duong Hung Anh

 F U N DA M E N TA L C O N C E P T S
Problem: Determine the moment of force(s) about a point (O).

Engineering Mechanics- Statics Le Duong Hung Anh

 F U N DA M E N TA L C O N C E P T S
Problem: Determine the moment of force(s) about a point (O).

Engineering Mechanics- Statics Le Duong Hung Anh

F U N D A M E N TA L C O N C E P T S
Moment of a force system about a point

mO ( F ) = r  F

Magnitude. ( )
mO F = M =  F .d

Direction: perpendicular to the plane that contains the force 𝐹

and 𝑟 .
The sense of direction: the right-hand rule

Engineering Mechanics- Statics Le Duong Hung Anh

FUNDAMENTAL CONCEPTS
M om e nt of a f or c e s ys t e m a bout a poi nt – Ca r t e s i a n Ve c t or

rx ry rz represent the x, y, z components of the position

vector drawn from point O to any point on the line of
action of the force.
Fx Fy Fz represent the x, y, z components of the force vector.

M O = (ry Fz − rz Fy )i + (rz Fx − rx Fz ) j + (rx Fy − ry Fx )k

.Since the force is parallel to the axis, there is no moment or tendency to cause turning about that axis.
.Mo is always be perpendicular to the shaded plane containing vectors r and F.
.Since the line action of force passes through the axis, there is no moment or tendency to cause turning about
that axis.

If a body is acted upon by a system of forces, the resultant can be written as

( M R )O =  ( r  F )

Engineering Mechanics- Statics Le Duong Hung Anh

F U N D A M E N TA L C O N C E P T S
Moment of a force system about a point – Cartesian Vector

Determine the moment produced by forces about point O (using scalar & non-scalar).

Position vector and Force vector.

A(0,0,6) m → 𝒓OA = 6k (m)
B (0,2.5,0) (m) → 𝒓OB = 2.5j (m)
 (0 − 0) (2.5 − 0) (0 − 6) 
F B = F u AB = 780  i + j + k = [300 j − 720k ]( N )
2 
 (0 − 0) 2
(2.5 − 0) 2
(0 − 6) 
Moment of FB about point O.
i j k
M O = rOA  FB = 0 0 6 = [−1800i]( N .m)
0 300 −720 Resultant
Moment of FC about point O. moment?
i j k
M O = rOA  FC = 0 0 6 = [1080i + 720 j ]( N .m)
120 −180 −360
Engineering Mechanics- Statics Le Duong Hung Anh
F U N D A M E N TA L C O N C E P T S & A X I O M S Y S T E M S O F S TAT I C S
Simplification of a Force and Couple System
Force System Equivalent System

Concurrent

R = F

Coplanar

 R = F

( )
 M R O =  mO ( F )

(M ) = m
R
O
O (F ) = R  d

Engineering Mechanics- Statics Le Duong Hung Anh

F U N D A M E N TA L C O N C E P T S & A X I O M S Y S T E M S O F S TAT I C S
Simplification of a Force and Couple System
Example: Replace the force and couple moment system acting on the beam in the below figure by an equivalent
resultant force, and find where its line of action intersects the beam, measured from point O.

Engineering Mechanics- Statics Le Duong Hung Anh

F U N D A M E N TA L C O N C E P T S & A X I O M S Y S T E M S O F S TAT I C S
Simplification of a Force and Couple System
Problem: Replace the loading system acting on the post by an equivalent resultant force and couple moment
at point B. Specify where the force acts, measured from B.

Engineering Mechanics- Statics Le Duong Hung Anh

F U N D A M E N TA L C O N C E P T S
M om e nt of a f or c e s ys t e m a bout a poi nt – Ca r t e s i a n Ve c t or

Homework: Determine the moment about point A of each of the three forces acting on the beam AB as
shown in the following systems.

Engineering Mechanics- Statics Le Duong Hung Anh

SUPPORTS & CONNECTIONS – REACTION FORCE
Problem: Determine the moment about point A of force system acting on the beam AB as shown in the following
models.

Engineering Mechanics- Statics Le Duong Hung Anh

SUPPORTS & CONNECTIONS – REACTION FORCE
Weightless Link

Problem: Analyze the reaction of the following systems.

Engineering Mechanics- Statics Le Duong Hung Anh

 P r o b l e m : Determine the reaction forces at the point A, B, O of the
following systems.

5
4

50 N

Engineering Mechanics- Statics Le Duong Hung Anh

Equilibrium of a rigid body
Problem: Determine the reactive components at the supports of the following systems.

Figure 1 Figure 2 Figure 3

Figure 4 Figure 5 Figure 6

Engineering Mechanics- Statics Le Duong Hung Anh

Equilibrium of a rigid body
Problem: Determine the components of reaction at the supports of the following systems.

Figure 1 Figure 2 Figure 3

Engineering Mechanics- Statics Le Duong Hung Anh

Structural Analysis
Problem: Determine the reative components at the supports and the force in each member of the truss as shown in
the following figures.

Figure 1 Figure 2 Figure 3

30°

Figure 4 Figure 5

Engineering Mechanics- Statics Le Duong Hung Anh

Structural Analysis
Problem: Determine the force in each member of the Problem: Determine the force in members EF, AF,
space truss and state if the members are in tension and DF of the space truss and state if the members
or compression are in tension or compression. The truss is supported
by short links at A, B, D, and E.

Engineering Mechanics- Statics Le Duong Hung Anh