MEKANIKA TEKNIK
OLEH:
PRAMONO
JURUSAN TEKNIK MESIN
FT – UNNES
Apa itu Mekanika?
Cabang ilmu fisika yang berbicara tentang
keadaan diam atau geraknya benda-benda
yang mengalami kerja atau aksi gaya
Mechanics
Rigid Bodies
Deformable Bodies
(Things that do not change shape) (Things that do change shape)
Statics
Dynamics
Fluids
Incompressible
Compressible
Review Sistem Satuan
• Four fundamental physical quantities. Length, Time, Mass, Force.
• We will work with two unit systems in static’s: SI & US Customary.
Bagaimana konversi dari SI ke US atau sebaliknya ?
SISTEM GAYA
SISTEM GAYA
SPACE (3D)
Fundamental Principles
• The parallelogram law for the addition of forces: Two
forces acting on a particle can be replaced by a single
force, called resultant, obtained by drawing the diagonal
of the parallelogram which has sides equal to the given
forces
f1+f2
f2
f1
• Parallelogram Law
Fundamental Principles (cont’)
• The principle of transmissibility: A force acting at a point
of a rigid body can be replaced by a force of the the same
magnitude and same direction, but acting on at a different
point on the line of action
f2
f1
f1 and f2 are equivalent if their
magnitudes are the same and the
object is rigid.
• Principle of Transmissibility
APPLICATION OF VECTOR
ADDITION
There are four
concurrent cable forces
acting on the bracket.
How do you determine
the resultant force acting
on the bracket ?
Addition of Vectors
• Trapezoid rule for vector addition
• Triangle rule for vector addition
C
B
C
B
• Law of cosines,
R 2 = P 2 + Q 2 − 2 PQ cos B
r r r
R = P+Q
• Law of sines,
sin A sin B sin C
=
=
Q
R
A
• Vector addition is commutative,
r r r r
P+Q = Q+ P
• Vector subtraction
Sample Problem
SOLUTION:
The two forces act on a bolt at A.
Determine their resultant.
• Trigonometric solution use the triangle rule for
vector addition in
conjunction with the law
of cosines and law of
sines to find the resultant.
• Trigonometric solution - Apply the triangle rule.
From the Law of Cosines,
R 2 = P 2 + Q 2 − 2 PQ cos B
= (40 N )2 + (60 N )2 − 2(40 N )(60 N ) cos155°
R = 97.73N
From the Law of Sines,
sin A sin B
=
Q
R
Q
sin A = sin B
R
60 N
= sin 155°
97.73N
A = 15.04°
α = 20° + A
α = 35.04°
ADDITION OF SEVERAL VECTORS
• Step 1 is to resolve each force
into its components
• Step 2 is to add all the x
components together and add all
the y components together.
These two totals become the
resultant vector.
• Step 3 is to find the magnitude
and angle of the resultant
vector.
Example of this process,
You can also represent a 2-D vector with a
magnitude and angle.
EXAMPLE
Given: Three concurrent forces
acting on a bracket.
Find:
The magnitude and
angle of the resultant
force.
Plan:
a) Resolve the forces in their x-y components.
b) Add the respective components to get the resultant vector.
c) Find magnitude and angle from the resultant components.
EXAMPLE (continued)
F1 = { 15 sin 40° i + 15 cos 40° j } kN
= { 9.642 i + 11.49 j } kN
F2 = { -(12/13)26 i + (5/13)26 j } kN
= { -24 i + 10 j } kN
F3 = { 36 cos 30° i – 36 sin 30° j } kN
= { 31.18 i – 18 j } kN
EXAMPLE (continued)
Summing up all the i and j components respectively, we get,
FR = { (9.642 – 24 + 31.18) i + (11.49 + 10 – 18) j } kN
= { 16.82 i + 3.49 j } kN
y
FR
FR = ((16.82)2 + (3.49)2)1/2 = 17.2 kN
φ = tan-1(3.49/16.82) = 11.7°
φ
x
Sample Problem
SOLUTION:
• Resolve each force into
rectangular components.
• Determine the components of the
resultant by adding the
corresponding force components.
Four forces act on bolt A as shown.
Determine the resultant of the
force on the bolt.
• Calculate the magnitude and
direction of the resultant.
Sample Problem (cont’)
SOLUTION:
• Resolve each force into rectangular components.
Sample Problem (cont’)
force
mag
r
F
1
r
F
2
r
F
3
r
F
4
150
80
x − comp
+ 129
y − comp
.9
+ 75 . 0
− 27 . 4
+ 75 . 2
110
0
100
+ 96 . 6
R x = +199.1
− 110
.0
− 25 . 9
R y = +14.3
• Determine the components of the resultant by adding the corresponding force
components.
• Calculate the magnitude and direction.
Ry
14.3 N
tan α =
=
α = 4.1°
Rx 199.1 N
14.3 N
R=
= 199.6 N
sin
α = 4.1°
MOMEN DAN KOPEL
Apa yang dipelajari sekarang ?
Mengetahui dan memahami maksud dari
momen gaya, momen kopel, dan cara
memindah gaya
Apa itu momen gaya ?
The moment of a force about a point provides a measure of the
tendency for rotation (sometimes called a torque).
MOMENT IN 2-D (continued)
In the 2-D case, the magnitude of the moment is
Mo = F d
As shown, d is the perpendicular distance from point O to the
line of action of the force.
In 2-D, the direction of MO is either clockwise or
counter-clockwise depending on the tendency for rotation.
Moment in 2-D
b
O
d
a
F
As shown, d is the
perpendicular distance
from point O to the line
of action of the force.
MO = F d
and the direction is
counter-clockwise.
Fy
a
b
F
Fx
O
Often it is easier to
determine MO by using the
components of F as shown.
MO = (FY a) – (FX b)
CCW = (+)
CW = (-)
Example 1
Given: A 40 N force is
applied to the wrench.
Find: The moment of the
force at O.
Plan: 1) Resolve the force
along x and y axes.
2) Determine MO using
scalar analysis.
Solution: + ↑ Fy = - 40 cos 20° N
+ → Fx = - 40 sin 20° N
+ MO = {-(40 cos 20°)(200) + (40 sin 20°)(30)}N·mm
= -7107 N·mm = - 7.11 N·m
EXAMPLE 2
Given: A 400 N force is
applied to the frame
and θ = 20°.
Find: The moment of the
force at A.
Plan:
1) Resolve the force along x and y axes.
2) Determine MA using scalar analysis.
EXAMPLE 2 (continued)
Solution
+ ↑ Fy = -400 cos 20° N
+ → Fx = -400 sin 20° N
+ MA = {(400 cos 20°)(2) + (400 sin 20°)(3)} N·m
= 1160 N·m
CONCEPT QUESTION
F = 10 N
1. What is the moment of the 10 N force
about point A (MA)?
A) 10 N·m
B) 30 N·m C) 13 N·m
D) (10/3) N·m
E) 7 N·m
d=3m
• A
2. If a force of magnitude F can be applied in four different 2-D
configurations (P,Q,R, & S), select the cases resulting in the
maximum and minimum torque values on the nut. (Max, Min).
A) (Q, P)
B) (R, S)
C) (P, R)
D) (Q, S)
S
R
P
Q
10 N
3m
P
2m
5N
3. Using the CCW direction as positive, the net moment of the
two forces about point P is
A) 10 N ·m
B) 20 N ·m
C) - 20 N ·m
D) 40 N ·m
E) - 40 N ·m
Apa itu momen kopel ?
Moment of a Couple
A couple is defined as two
parallel forces with the same
magnitude but opposite in
direction separated by a
perpendicular distance d.
The moment of a couple is defined as
MO = F d (using a scalar analysis) or as
MO = r × F (using a vector analysis).
Here r is any position vector from the line of action of
–F to the line of action of F.
Problem Solving
A
B
A torque or moment of 12 N · m is required to rotate the wheel.
Which one of the two grips of the wheel above will require less
force to rotate the wheel?
Problem Solving (2-D)
A
B
M=Fd
M=Fd
12 = F 0.4
12 = F 0.3
F = 30 N
F = 40 N
PROBLEM SOLVING - SCALAR
Given: Two couples act on the
beam. The resultant couple is
zero.
Find: The magnitudes of the forces
P and F and the distance d.
PLAN:
1) Use definition of a couple to find P and F.
2) Resolve the 300 N force in x and y directions.
3) Determine the net moment.
4) Equate the net moment to zero to find d.
Solution:
From the definition of a
couple
P = 500 N and
F = 300 N.
Resolve the 300 N force into vertical and horizontal
components. The vertical component is (300 cos 30º) N and
the horizontal component is (300 sin 30º) N.
It was given that the net moment equals zero. So
+ ΣM = - (500)(2) + (300 cos 30º)(d) + (300 sin 30º)(0.2) = 0
Now solve this equation for d.
d = (1000 – 60 sin 30º) / (300 cos 30º) = 3.96 m
CONCEPT QUESTION
1. In statics, a couple is defined as __________ separated by a
perpendicular distance.
A) two forces in the same direction.
B) two forces of equal magnitude.
C) two forces of equal magnitude acting in the same direction.
D) two forces of equal magnitude acting in opposite directions.
2. F1 and F2 form a couple. The moment
of the couple is given by ____ .
A) r1 × F1
B) r2 × F1
C) F2 × r1
D) r2 × F2
F1
r1
F2
r2
3. A couple is applied to the beam as shown. Its moment
equals _____ N·m.
50 N
A) 50
B) 60
C) 80
D) 100
1m
2m
5
3
4
Apa itu memindah gaya ?
Several forces and a
couple moment are
acting on this vertical
section of an I-beam.
Can you replace them
with just one force and
one couple moment at
point O that will have
the same external
effect? If yes, how will
you do that?
AN EQUIVALENT SYSTEM
=
When a number of forces and couple moments are acting on a
body, it is easier to understand their overall effect on the body if
they are combined into a single force and couple moment having
the same external effect
The two force and couple systems are called equivalent systems
since they have the same external effect on the body.
Equivalent Force – Couple Systems
Moving a force from A to O, when both points are on the
vectors’ line of action, does not change the external effect.
Hence, a force vector is called a sliding vector. (But the
internal effect of the force on the body does depend on
where the force is applied).
Equivalent Force – Couple Systems
Moving a force from point A to O (as shown above) requires
creating an additional couple moment. Since this new couple
moment is a “free” vector, it can be applied at any point P on the
body.
Equivalent Force – Couple Systems
If the force system lies in the x-y plane (the 2-D case), then the
reduced equivalent system can be obtained using the following
three scalar equations.
Problem Solving (2-D)
Given: A 2-D force and couple
system as shown.
Find: The equivalent resultant force
and couple moment acting at
A and then the equivalent
single force location along
the beam AB.
Plan:
1) Sum all the x and y components of the forces to find FRA.
2) Find and sum all the moments resulting from moving each
force to A.
3) Shift the FRA to a distance d such that d = MRA/FRy
Problem Solving (2-D)
+ → ΣFRx = 25 + 35 sin 30°
= 42.5 N
d
FR
+ ↑ ΣFRy = - 20 - 35 cos 30°
= - 50.31 N
+
MRA
= - 35 cos30° (0.2)
- 20(0.6) + 25(0.3)
= - 10.56 N.m
FR = ( 42.52 + (-50.31)2 )1/2 = 65.9 N
θ = tan-1 ( 50.31/42.5) = 49.8 ° (Kw IV)
The equivalent single force FR can be located on the
beam AB at a distance d measured from A.
d = MRA/FRy = - 10.56/(-50.31) = 0.21 m.
KESEIMBANGAN
PARTIKEL (2D)
Equilibrium of a Particle (2-D)
Today’s Objectives:
Students will be able to :
a) Draw a free body diagram
(FBD), and,
b) Apply equations of
equilibrium to solve a 2-D
problem.
For a given cable
strength, what is the
maximum weight
that can be lifted ?
Apa pentingnya mekanika (statik) /
keseimbangan ?
Apa perbedaan partikel dan benda tegar?
• Particle: A very small amount of matter which
may be assumed to occupy a single point in
space.
• Rigid body: A combination of a large number
of particles occupying fixed position with
respect to each other.
Apa perbedaan Partikel dan Benda Tegar ?
Partikel:
Mempunyai suatu
massa namun
ukurannya dapat
diabaikan, sehingga
geometri benda tidak
akan terlibat dalam
analisis masalah
Benda Tegar:
Kombinasi sejumlah
partikel yang mana
semua partikel
berada pada suatu
jarak tetap terhadap
satu dengan yang lain
Contoh Partikel
Contoh Benda Tegar
THE WHAT, WHY AND HOW OF A
FREE BODY DIAGRAM (FBD)
Free Body Diagrams are one of the most important things for
you to know how to draw and use.
What ? - It is a drawing that shows
all external forces acting on the
particle.
Why ? - It helps you write the
equations of equilibrium used to
solve for the unknowns (usually
forces or angles).
How ?
1. Imagine the particle to be isolated or cut free from its
surroundings.
2. Show all the forces that act on the particle.
Active forces: They want to move the particle.
Reactive forces: They tend to resist the motion.
3. Identify each force and show all known magnitudes
and directions. Show all unknown magnitudes and /
or directions as variables .
A
Note : Engine mass = 250 Kg
FBD at A
Free Body Diagram (FBD) (2-D)
Equations of Equilibrium (2-D)
Since particle A is in equilibrium,
the net force at A is zero.
So FAB + FAD + FAC = 0
or Σ F = 0
In general, for a particle in equilibrium, Σ F = 0 or
ΣFx i + ΣFy j = 0 = 0 i + 0 j (A vector equation)
Or, written in a scalar form,
These are two scalar equations of equilibrium (EofE). They
can be used to solve for up to two unknowns.
ΣFx = 0 and Σ Fy = 0
EXAMPLE
Note : Engine mass = 250 Kg
FBD at A
Write the scalar EofE:
+ → Σ Fx = TB cos 30º – TD = 0
+ ↑ ΣFy = TB sin 30º – 2.452 kN = 0
Solving the second equation gives: TB = 4.90 kN
From the first equation, we get: TD = 4.25 kN
Problem Solving (2-D)
F = 600 N
Given: The car is towed at constant
speed by the 600 N force
and the angle θ is 25°.
θ=
25o
Find: The forces in the ropes AB
and AC.
Plan:
1. Draw a FBD for point A.
2. Apply the EofE to solve for the forces
in ropes AB and AC.
Problem Solving (2-D)
F = 600 N
600 N
FBD at point A
A
θ=
25o
25°
FAB
30°
FAC
Applying the scalar EofE at A, we get;
+ → ∑Fx = FAC cos 30° – FAB cos 25° = 0
+ → ∑Fy = -FAC sin 30° – FAB sin 25° + 600 = 0
Solving the above equations, we get;
FAB = 634 N
FAC = 664 N
Example
SOLUTION:
• Construct a free-body diagram for the
particle at the junction of the rope and
cable.
• Determine the unknown force
magnitudes.
In a ship-unloading operation, a
15.6 kN automobile is supported by
a cable. A rope is tied to the cable
and pulled to center the automobile
over its intended position. What is
the tension in the rope?
EXAMPLE
Given: Sack A weighs 20 N.
and geometry is as
shown.
Find: Forces in the cables and
weight of sack B.
Plan:
1. Draw a FBD for Point E.
2. Apply EofE at Point E to solve
for the unknowns (TEG & TEC).
3. Repeat this process at C.
EXAMPLE (continued)
A FBD at E should look like the one
to the left. Note the assumed
directions for the two cable tensions.
The scalar EofE are:
+ → Σ Fx = TEG sin 30º – TEC cos 45º = 0
+ ↑ Σ Fy = TEG cos 30º – TEC sin 45º – 20 N = 0
Solving these two simultaneous equations for the
two unknowns yields:
TEC = 38.6 N
TEG = 54.6 N
EXAMPLE (continued)
Now move on to ring C.
A FBD for C should look
like the one to the left.
The scalar EofE are:
+ → Σ Fx = 38.64 cos 45° – (4/5) TCD = 0
+ ↑ Σ Fy = (3/5) TCD + 38.64 sin 45° – WB = 0
Solving the first equation and then the second yields
TCD = 34.2 N
and
WB = 47.8 N .
READING QUIZ
1) When a particle is in equilibrium, the sum of forces acting
on it equals ___ . (Choose the most appropriate answer)
A) a constant
B) a positive number C) zero
D) a negative number E) an integer.
2) For a frictionless pulley and cable, tensions in the cable
(T1 and T2) are related as _____ .
A) T1 > T2
B) T1 = T2
C) T1 < T2
D) T1 = T2 sin θ
ATTENTION QUIZ
3. Using this FBD of Point C, the sum of
forces in the x-direction (Σ FX) is ___ .
Use a sign convention of + → .
A) F2 sin 50° – 20 = 0
B) F2 cos 50° – 20 = 0
C) F2 sin 50° – F1 = 0
D) F2 cos 50° + 20 = 0
F2
20 N
50°
C
F1
SOAL TANTANGAN
a
b
C
10 m
B
5m
P
A
80 kg
40 kg
Jika b = 4 m,
tentukan harga P dan jarak a
KESEIMBANGAN
BENDA TEGAR
Apa Beda Partikel dengan Benda Tegar ?
In contrast to the forces on a particle, the
forces on a rigid-body are not usually
concurrent and may cause rotation of the
body (due to the moments created by the
forces).
Forces on a particle
For a rigid body to be in equilibrium, the
net force as well as the net moment
about any arbitrary point O must be
equal to zero.
∑ F = 0 and ∑ MO = 0
Forces on a rigid body
Benda Tegar Biasanya Memiliki Tumpuan
Benda Tegar Biasanya Memiliki Tumpuan
Macam-macam Tumpuan dan Reaksinya
Contoh Menggambar FBD nya
Idealized model
Free body diagram
Lho kok ada beban yang segiempat, apa itu?
Beban Terdistribusi
Mencari Gaya Resultan pada Beban Terdistribusi
• Mencari titik berat dari beban terdistribusi
• Gaya resultan sama dengan luasan dari beban
terdistribusi
• Gaya resultan terletak pada titik berat beban
terdisribusi
Kalo beban terdistribusinya berbentuk segitiga ?
FR
100 N/m
12 m
1. FR = ____________
x
2. x = __________.
A) 12 N
B) 100 N
A) 3 m
B) 4 m
C) 600 N
D) 1200 N
C) 6 m
D) 8 m
Prosedur Menyelesaikan Soal
•
•
•
•
Gambar FBD dari soal
Jangan lupa kasih perjanjian tandanya
Gambar gaya reaksi yang ada
Kalo ada beban terdistribusi, cari dulu besar
gaya resultan, dan posisinya
• Hitung besar gaya reaksi di tumpuan,
menggunakan
∑ Fx = 0
∑ Fy = 0
titik O itu titik apa? Yang mana?
∑ Mo = 0
Contoh Soal 1
Given: Weight of the boom =
125 lb, the center of
mass is at G, and the
load = 600 lb.
Find: Support reactions at A
and B.
Plan:
1. Put the x and y axes in the horizontal and vertical directions,
respectively.
2. Draw a complete FBD of the boom.
3. Apply the EofE to solve for the unknowns.
Contoh Soal 1 (Jawaban)
FBD of the boom:
AY
AX
A
1 ft
1 ft
3 ft
B
40°
FB
5 ft
D
G
125 lb
600 lb
+ ∑MA = - 125 ∗ 4 - 600 ∗ 9 + FB sin 40° ∗ 1 + FB cos 40° ∗ 1 = 0
FB = 4188 lb or 4190 lb
→ + ∑FX = AX + 4188 cos 40° = 0;
AX = – 3210 lb
↑ + ∑FY = AY + 4188 sin 40° – 125 – 600 = 0;
AY = – 1970 lb
Contoh Soal 2
SOLUTION:
• Create a free-body diagram for the crane.
• Determine B by solving the equation for
the sum of the moments of all forces
about A. Note there will be no
contribution from the unknown
reactions at A.
A fixed crane has a mass of 1000 kg
and is used to lift a 2400 kg crate. It
is held in place by a pin at A and a
rocker at B. The center of gravity of
the crane is located at G.
Determine the components of the
reactions at A and B.
• Determine the reactions at A by
solving the equations for the sum of
all horizontal force components and
all vertical force components.
• Check the values obtained for the
reactions by verifying that the sum of
the moments about B of all forces is
zero.
Contoh Soal 2 (jawaban)
• Determine B by solving the equation for the
sum of the moments of all forces about A.
∑ M A = 0 : + B(1.5m ) − 9.81 kN(2m )
− 23.5 kN(6m ) = 0
B = +107.1 kN
• Create the free-body diagram.
• Determine the reactions at A by solving the
equations for the sum of all horizontal forces
and all vertical forces.
∑ Fx = 0 : Ax + B = 0
Ax = −107.1 kN
∑ Fy = 0 : Ay − 9.81 kN − 23.5 kN = 0
Ay = +33.3 kN
• Check the values obtained.
Contoh Soal 3
ΣM A = 0 = .400 m ( B y ) − 120 N (.275 m) − 225 N (.500 m)
+
− 120 N (.275 m) − 225 N (.500 m)
By =
− .400 m
B y = 363.75 N ↑
Σ+Fy = 0 = Ay − 120 N + 363.75 N − 225 N
Ay = −18.75 N
Ay = 18.75 N ↓
Contoh Soal 4
Given: The loading on the beam as
shown.
Find: Support reactions at A and B.
Contoh Soal 4 (jawaban)
SOLUTION:
• Taking entire beam as a free-body,
determine reactions at supports.
∑ F y =0 :
∑MA = 0:
Ay − 90 kN − 54 kN + 117 kN − 54 kN = 0
D(7.2 m) − (90 kN )(1.8 m) − (54 kN )(4.2 m)
− (54 kN )(8.4 m) = 0
D = 117 kN
Ay = 81 kN
Contoh Soal 5
Tentukan Reaksi di A dan B
Soal Tantangan
Given: The loading on the beam as shown.
Find: Reaction at B and A
Soal Tantangan (2)
Tentukan Reaksi di A dan C
BEAM
GAYA INTERNAL,
DIAGRAM GAYA GESER
DAN MOMEN
Definisi Beam
• Beam - structural member designed to support
loads applied at various points along its length.
• Beam can be subjected to concentrated loads or
distributed loads or combination of both.
• Beam design is two-step process:
1) determine shearing forces and bending
moments produced by applied loads
2) select cross-section best suited to resist
shearing forces and bending moments
Apa itu Gaya Internal ?
Gaya Internal : gaya yang mengikat bersama berbagai
bagian struktur sehingga struktur tersebut menjadi kokoh
• Straight two-force member AB is in
equilibrium under application of F and
-F.
• Internal forces equivalent to F and -F are
required for equilibrium of free-bodies AC
and CB.
Reaksi pada Beam
• Beams are classified according to way in which they are
supported.
• Reactions at beam supports are determinate if they
involve only three unknowns. Otherwise, they are
statically indeterminate.
Gaya Geser dan Momen pada Beam
• Wish to determine bending moment
and shearing force at any point in a
beam subjected to concentrated and
distributed loads.
• Determine reactions at supports by
treating whole beam as free-body.
• Cut beam at C and draw free-body
diagrams for AC and CB. By
definition, positive sense for internal
force-couple systems are as shown.
• From equilibrium considerations,
determine M and V or M’ and V’.
Diagram Gaya Geser dan Momen pada Beam
• Variation of shear and bending
moment along beam may be
plotted.
• Determine reactions at
supports.
• Cut beam at C and consider
member AC,
V = + P 2 M = + Px 2
• Cut beam at E and consider
member EB,
V = − P 2 M = + P(L − x ) 2
• For a beam subjected to
concentrated loads, shear is
constant between loading points
and moment varies linearly.
Contoh Soal 1
SOLUTION:
• Taking entire beam as a free-body,
calculate reactions at B and D.
Draw the shear and bending moment
diagrams for the beam and loading
shown.
• Find equivalent internal force-couple
systems for free-bodies formed by
cutting beam on either side of load
application points.
• Plot results.
Contoh Soal 1 (jawaban)
• Plot results.
Note that shear is of constant value
between concentrated loads and
bending moment varies linearly.
Contoh Soal 2
Given: A beam is supported
by a hinge at A, a roller at
B. Force applied at C.
Moment applied at D.
Find: Draw the shear and
bending moment diagrams
Plan:
a) Draw a FBD of the beam.
b) Calculate support reactions.
c) Find equivalent internal force-couple systems for free-bodies formed
by cutting beam on either side of load application points.
Contoh Soal 2 (jawaban)
Ay
By
ΣMB = 0
ΣFy = 0
- Ay(20) + 0.6(10) – 4 = 0
Ay + By – 600 = 0
20Ay = 6 - 4
0.1 + By – 0.6 = 0
Ay = 0.1 kip
By = 0.5 kip
Contoh Soal 2 (jawaban)
Potongan 1-1
2
1
3
V
M
1
0.1 kip
2
3
x
0.5 kip
0.1 kip
0 ≤ x < 10 ft
ΣM1-1 = 0
ΣF1-1 = 0
M – 0.1(x) = 0
0.1 – V = 0
M = 0.1(x)
V = 0.1 kip
x = 0 ft
Æ MA = 0.1(0) = 0 kip-ft
x = 10 ft Æ MC = 0.1(10) = 1 kip-ft
x = 0 ft
Æ VA = 0.1 kip
x = 10 ft Æ VC = 0.1 kip
Contoh Soal 2 (jawaban)
Potongan 2-2
2
1
3
V
M
1
2
0.1 kip
3
0.5 kip
0.1 kip
x
10 ≤ x < 15 ft
ΣM2-2 = 0
ΣF2-2 = 0
M – 0.1(x) + 0.6(x – 10) = 0
0.1 – 0.6 – V = 0
M – 0.1(x) + 0.6(x) – 6 = 0
V = 0.1 – 0.6 kip
M = - 0.5(x) + 6
V = - 0.5 kip
x = 10 ft Æ MC = - 0.5(10) + 6 = 1 kip-ft
x = 10 ft Æ VC = - 0.5 kip
x = 15 ft Æ MD = - 0.5(15) + 6 = - 1.5 kip-ft x = 15 ft Æ VD = - 0.5 kip
Contoh Soal 2 (jawaban)
Potongan 3-3
2
1
3
V
M
1
2
3
0.1 kip
0.5 kip
0.1 kip
x
15 ≤ x < 20 ft
ΣM3-3 = 0
ΣF3-3 = 0
M – 0.1(x) + 0.6(x – 10) – 4 = 0
0.1 – 0.6 – V = 0
M – 0.1(x) + 0.6(x) – 6 – 4 = 0
V = 0.1 – 0.6 kip
M = - 0.5(x) + 10
V = - 0.5 kip
x = 15 ft Æ MD = - 0.5(15) + 10 = 2.5 kip-ft x = 15 ft Æ VD = - 0.5 kip
x = 20 ft Æ MB = - 0.5(20) + 10 = 0 kip-ft
x = 20 ft Æ VD = - 0.5 kip
Contoh Soal 2 (jawaban)
Diagram Geser dan Momen
ΣM1-1 = 0
M = 0.1(x)
MA = 0 kip-ft
MC = 1 kip-ft
ΣF1-1 = 0
V = 0.1 kip
VA = 0.1 kip
VC = 0.1 kip
ΣM2-2 = 0
M = - 0.5(x) + 6
MC = 1 kip-ft
MD = - 1.5 kip-ft
ΣF2-2 = 0
V = - 0.5 kip
VC = - 0.5 kip
VD = - 0.5 kip
ΣM3-3 = 0
M = - 0.5(x) + 10
MD = 2.5 kip-ft
MB = 0 kip-ft
ΣF3-3 = 0
V = - 0.5 kip
VD = - 0.5 kip
VB = - 0.5 kip
Contoh Soal 3
SOLUTION:
• Taking entire beam as free-body,
calculate reactions at A and B.
• Determine equivalent internal forcecouple systems at sections cut within
segments AC, CD, and DB.
Draw the shear and bending moment
diagrams for the beam AB. The
distributed load of 7200 N/m. extends
over 0.3 m of the beam, from A to C,
and the 1800 N load is applied at E.
• Plot results.
Contoh Soal 3 (jawaban)
SOLUTION:
• Taking entire beam as a free-body, calculate
reactions at A and B.
∑MA = 0:
B y (0.8 m) − (2160 N )(0.15 m) − (1800 N )(0.55 m) = 0
B y = 1642 N
∑MB = 0:
(2160 N)(0.65 m) + (1800 N)(0.25 m) − A(0.8 m) = 0
A = 2318 N
∑ Fx = 0 :
Bx = 0
• Note: The 1800 N load at E may be replaced by
a 1800 N force and 180 Nm. couple at D.
Contoh Soal 3 (jawaban)
• Evaluate equivalent internal force-couple systems
at sections cut within segments AC, CD, and DB.
From A to C:
∑ Fy = 0 : 2318 − 7200 x − V = 0
V = 2318 − 7200 x
1
∑ M 1 = 0 : − 2318 x + 7200 x( 2 x ) + M = 0
M = 2318 x − 3600 x 2
From C to D:
∑ Fy = 0 : 2318 − 2160 − V = 0
V = 158 N
∑ M 2 = 0 :− 2318 x + 2160( x − 0.15) + M = 0
M = (324 + 158 x ) N ⋅ m
Contoh Soal 3 (jawaban)
• Evaluate equivalent internal force-couple
systems at sections cut within segments AC,
CD, and DB.
From D to B:
∑ Fy = 0 : 2318 − 2160 − 1800 − V = 0
V = −1642 N
∑M2 = 0:
− 2318 x + 2160( x − 0.15) − 180 + 1800( x − 0.45) + M = 0
M = (1314 − 1642 x ) N ⋅ m
Contoh Soal 3 (jawaban)
• Plot results.
From A to C:
V = 3218 − 7200 x
M = 3218 x − 20 x 2
From C to D:
V = 158 N
M = (324 + 158 x ) N ⋅ m
From D to B:
V = −1642 N
M = (1314 − 1642 x ) N ⋅ m
Contoh Soal 4
Draw the shear and bending-moment diagrams
for the beam and loading shown.
Contoh Soal 4 (jawaban)
What’s Stopping You?
Mu
F
n
Friction
Hard to Live With It, Can’t Live Without It
Coefficient of Friction
Consider 2 Types of Friction
Fs
Force of Static Friction
This value represents the relative force necessary
to make an object move
Fk
Force of Kinetic Friction
This value represents the relative force necessary
to keep an object moving at a constant rate
Resisting Motion
Frictional Force
Fs
Fs ≥ Fk
Max
Fk
Fs ≥ Fk
Static
Region
Kinetic Region
Force Causing the Object to Move
Frictional Forces Occur When
Materials are in N Contact
F
fs
M1
Surfaces in
Contact
Spring Scale
W
F = Force Causing Motion (Pull on Scale)
Fs = Force of Static Friction (Resists Motion)
N = Force Normal Holds Surfaces in Contact
W = Weight of Object ( Mass x Gravity)
Friction is a Force That
N
Resists Motion
F
fs
M1
Surfaces in
Contact
Spring Scale
W
The Pink Block M1 Will not Move Until the
Force F (Pull on the scale ) Exceeds the
Force of Static Friction fs.
The Relative Force of Static Friction Between
2 Objects is Expressed as the N Quotient of
the Force (F) Required to
Move the
Object
F
fs
M1
Surfaces in
Contact
Spring Scale
W
Divided by the Weight W of the Object
This is Called the Coefficient of
Friction
F Coefficient of Static
µs =
N
Friction
W
F
µ
fs
M1
Surfaces in
Contact
Spring Scale
W
µs
F
W
= Coefficient of Friction
= Force Required to Cause Motion
= Weight of Object
Using the Gears-IDS Battery to
Calculate The µ s Static
F
µs =
W
Coefficient of Friction
Record the Maximum Force (F)
(Before the Battery Begins to Move)
Maximum Force F = 110 g
F
µs =
W
Record the Weight
(W ) of the Battery
F
µs =
W
580 g
The Coefficient of Static Friction Between
the Wood Desktop and the Plastic Battery
is Described Algebraically:
F
µs =
W
µs
= .190
= 110 g
= 580 g
The Coefficient of Kinetic Friction Can
be Found Using the Same Technique
F
µk =
W
Record the Force Required to Move
the Battery at a Constant Rate
Coefficients of Friction Between Various
Surfaces Have Significant Impact on the
Design and Construction of Mechanisms.
The Design of Competitive
Mechanisms with Wheels Requires
Extensive Testing in Order to
Optimize the Torque, Speed and
Traction (Friction) of the Drive
Systems.
The End