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FORCE SYSTEMS

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F

F

W

P

P

2

(tvid - 2a) ,

(tvid - 2b)

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F

For your own good, learn this topic well, my friend!

2.1 Importance of the Subject :

  • Force is a vector quantity.
  • Force is the most important concept in the Statics course.
  • All operations performed for vectors can also be performed for forces.
  • There are also vector definitions and operations that can be performed with forces. Learning these is very important for the purposes of the course.
  • If we thoroughly understand the operations in this section, we will be able to solve static equilibrium problems much more easily and with understanding.

2. Force Systems

Figure 2.1

Figure 2.2

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1- External Effect: The external effect of the force is to try to move the object.

The force applied to an object produces two different effects on the object:

2.3 Effects of Force

2- Internal Effect: The internal effect of forces is to create stress and deformations, forcing the object to deform and damage. This is especially evident in balanced force systems. The calculation of stresses and deformations is the subject of the Strength course.

  • Force is the effect applied to an object from outside and forcing it to move.
  • Force is a vector quantity that has a magnitude, direction, and point of application.

2.2 What is Force?

If the force is balanced, the object is stationary (or moves at a constant speed) and the calculation of other forces that provide balance is the subject of the Statics course. (If the force is not balanced, the object moves with acceleration, in which case the motion is examined within the scope of the Dynamics course.)

2. Force Systems

Figure 2.3

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a-) According to the area it affects: If the dimensions of the area where the force is applied are very small compared to the dimensions of the entire object, the force is called a «Single force". If the area where the force is applied is too large to be neglected, it is called a "distributed Force".

Single Force

2.4 Classification of Forces

2. Force Systems

Distributed Force

Figure 2.4

Figure 2.5

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b-) According to Application Methods:

2. Force Systems

Concurrent Forces

Coplanar Forces

Colinear Forces

Although their lines are different, they are forces located in the same plane.

Figure 2.6

Figure 2.7

Paralel Forces

Figure 2.8

Figure 2.9

These are forces whose extensions meet at the same point.

These are forces located on the same line.

Forces that are parallel to each other even though they are not in the same plane.

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c-) Contact and Friction Forces

2. Force Systems

Let's consider two objects in contact:

 

 

 

 

 

 

 

 

In topic 7, the concept of friction will be explained in detail and examples will be solved.

Figure 2.10

(a)

(b)

tangent

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  • These forces are along the directions of the ropes, cables and ropes whose weights are neglected and away from them.
  • If the weights are not neglected, these forces are in the tangent direction at the point of application.
  • Ropes, cables and ropes carry forces only when they are tense.
  • In other words, these objects carry only tensile loads, not compressive loads.

Heavy Cable

d-) Forces in Ropes and Cables

Lightweight Cable (Weight is ignored.)

2. Force Systems

Figure 2.11

Figure 2.12

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Pulleys are grooved rollers used to change the direction of ropes or belts and to obtain high output force with low input force. In the frictionless condition, the tension forces at both ends of the belt attached to the pulley are equal.

e-) Forces in Pulleys

Pulley

(In frictionless pulley)

2. Force Systems

Figure 2.13

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Spring Forces,

  • increase as the spring lengthens,
  • are always in the direction of the spring,
  • are in the direction that will try to return the spring to its original position.

f-) Forces on Springs

2. Force Systems

 

 

 

Figure 2.14

(a)

(b)

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2.5 Vectorial Definition of a Force in Three Dimensions

a-) If the direction of the force is given by two angles;

2. Force Systems

 

 

 

 

 

Figure 2.15

(2.1.a)

(2.1.b)

(2.1.c)

(2.1.d)

(2.2)

(2.3)

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b-) If the coordinates of two points on the line of action of the force are given;

 

If you don't understand, take a look at example 1.3.

2. Force Systems

 

 

 

 

 

Figure 2.16

(eq.1.10)

(eq.1.11)

(eq.1.12)

(2.4.a)

(2.4.b)

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2.6 Rotating Effect of Force: MOMENT

  • A force forces an object to rotate just as it forces it to translate.
  • This rotating effect is called a moment.
  • Moment is a vector quantity.
  • Its direction is perpendicular to the plane of rotation.
  • Its direction is found with the right-hand rule.

2. Force Systems

 

F

(It will be explained later.)

Figure 2.17

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  • Go to the door closest to you..
  • Open the door all the way..Now;
  • 1-Try to close the door by pushing from

the closest point to the hinges,

  • 2-Try to close it by pushing from the middle,
  • 3-Try to close it by pushing from the very end.
  • Which one did you have difficulty with?
  • Of course, you had difficulty when you pushed from the closest point to the hinges.
  • Maybe you couldn't close it. Think about why..

a- Feel the effect of the moment.

2. Force Systems

  • The closing of the door is related to the rotation effect of the pushing force, that is, the moment. In order for it to close, the moment that your pushing force will create must exceed a certain intensity.
  • The intensity of the moment is directly proportional to the force (F) and the force arm (d).
  • The force arm (d) is the perpendicular distance of the force from the rotation axis.
  • In this case, the intensity of the moment = Force x Force arm… in other words, … M = Fd … is defined as.
  • The farther you are from the rotation axis (hinges), the less force you apply for the same moment. Because the force arm «d» has increased. A lower F is sufficient for the same M. This means that you apply less force and have less difficulty.

Figure 2.18

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2. Force Systems

GIVE ME A LEVER AND I'LL MOVE THE WORLD.

OK

IT'S NOT POSSIBLE WITH THIS,

IT'S BROKEN

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2.7 Moment of a force about a point (important)

The moment of a force about a point is equal to vector multiplication of the position vector drawn from the point where moment is taken to any point on force and the force.

According to the figure 2.19 on the side; the moment of force F on line CD with respect to point A is found as follows:

 

point at which moment is taken

any point on line of force.

Force

 

Tip 2.4 ) If we multiply the vertical distance (d) by the magnitude of F, we only obtain the magnitude of the moment (M=Fd). However, if we perform vector multiplication, this operation is not necessary. Because when the vector expression of the moment is known, its magnitude can also be calculated.

 

2. Force Systems

 

(2.5)

Force line

α

A

B

 

d

 

 

.

.

 

k plane

 

C

D

 

Figurel 2.19

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Magnitude of Moment:

or

After finding

 

If the perpendicular distance is known :

 

 

2. Force Systems

Magnitude:

 

 

 

Figure 2.20

We repeat.: The moment of the force F relative to point O is equal to the vector product the position vector drawn from the point where the moment is taken (O) to any point on the force line (A, C, D or E) and the force.

(2.6)

(2.8)

(2.7)

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2.8- Varignon Theorem

This theorem states that: The moment of a force about a point is equal to the sum of the moments of the components of that force about the same point.

 

 

If the perpendicular distances are known, the magnitude of the moment is:

2. Force Systems

(The moment of P is taken as negative because the turning effect of P is in the opposite direction to that of Q.)

 

Figure2.21

(2.9)

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Example 2.1: Find the moment of the force F, which has a magnitude of 130 N and is directed from A to B, passing through the points A(3,8,1) and B(7,–4,4), about the point C(-1,1,1).

2. Force Systems

 

 

 

 

 

 

 

point where moment is taken

any point on line of force.

Solution:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Unit vector:

Figure 2.22

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2. Force Systems

Question 2.1 (*)

Vectorically calculate the moment of the 110kN force parallel to the y-z plane about point O.

 

Answer:

Figure 2.23

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2.9 Moment of a Force About an Axis

First, the moment of the force about a point on the axis is taken

and then the projection of this moment about the axis is found.

This projection vector is defined as the moment of the force about the axis.

  • If the extension of the force's direction coincides with the axis, the moment of the force about that axis will be zero.
  • This process is very useful in some 3-dimensional problems. It allows us to not include forces not asked in the question in the operations and to find a force that is asked directly with a single operation. An example will be given later.

2. Force Systems

Δ

 

For example, we are looking for the moment of force F with respect to axis Δ. Our process steps should be as follows:

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 2.24

(eq.2.5)

(2.10)

(2.11)

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O(0,0,0); A(0,3,12); B (5,,4,0); C (-3,5,4)

 

 

 

unit vector in the F direction:

Example 2.2

A force of F= 4 kN is applied to the BA rope attached to the pole in the figure. Find the moment of the force F about the OC axis.

First, let's take the moment about point O, which is any point on the axis:

Çözüm:

 

2. Force Systems

Figure 2.25

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θ =39.530

=(-14.72)x(-0.42)+18.4x0.71+ (-4.56)x(0.56)

 

Unit vector in the OC direction:

The magnitude of the moment about the OC axis:

(found by scalar multiplication.)

Moment vector about OC axis:

O

 

C

θ

 

 

 

2. Force Systems

Figure 2.26

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Question 2.2 (*)

 

2. Force Systems

Answers:

 

a-)

b-)

 

;

Figure 2.27

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2.10 Couple

If the rotation effect on an object is caused by two forces of equal intensity, opposite direction and parallel to each other, these forces are said to form a couple.

When opening or closing a large valve, we form a couple with our two hands.

Moment of a couple; is the sum of the moments of each of the forces about the midpoint.For example: If the valve radius is r, the couple moment is:

M = Fr + Fr = 2Fr.

A careless driver who does not apply coupling to the steering wheel.

2. Force Systems

F

F

r

M

Figure 2.28

Figure 2.29

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A force is transfered from one point to another with its moment.

In this way, the rotation effect is also preserved.

A moment is transferred from one point to another as it is. Because its effect does not change in this way.

2.11 Transfering a force from one point to another:

2.12 Transferring a moment from one point to another:

a force transfered from B to A

A moment transferred from B to A

If you pay attention, what is important in the transportation process is to preserve the translational or rotational effect of the force or moment on the object.

2. Force Systems

Figure2.30

Figure2.31

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2.13 Reduction of the System of Forces, and Resultants

 

 

2. Force Systems

 

(Vector sum of all singular forces)

(Moments of all singular forces about the reduced point + Singular moments) = Resultant Force

 

 

Figure 2.32

(2.12)

(2.13)

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The effect of the resultant alone on an object is equal to the effect of the components applied simultaneously.

2. Force Systems

Figure2.33

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Example 2.3:

Reduce the system of forces in the figure to point O.

2. Force Systems

Solution: Asking the question in a different way: Find a resultant force and resultant moment that will create the same effect at point O.

 

 

 

 

 

 

Figure 2.34

Figure 2.35

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Resultant Force:

Resultant Moment:

2. Force Systems

 

 

 

 

 

 

 

Figure 2.36

Figure 2.37

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2.14 Screw concept:

When tightening a screw with a screwdriver, we apply 2 types of movement and load at the same time.

1- We turn the screwdriver around its axis. (At this time, we apply torque to the screw).

2- At the same time, we press the screwdriver in the direction of its axis. (At this time, we apply compression force to the screw.)

 

Positive Force Screw (Tightening)

Negative Force Screw (Disassembly)

2. Force Systems

 

 

Figure 2.38

Figure 2.39

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or

=

a-) Detection of Screw Forming Condition:

If a system of forces is reduced to a point and the resultant force and resultant moment at the reduced point are on the same line, they are said to form a screw.

Question-1: How can we understand whether the resultant force and resultant moment form a screw?

 

2. Force Systems

When a system of forces is reduced to a point, If the resultant force and resultant moment are on the same line, they are said to form a screw.

Figure 2.40

Positive screw

Negative screw

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b-) Determination of the Reduction Status to Screw:

 

Question-2: Accordingly, what is the reduction condition for screw?

 

Question-3: How do we know whether M is perpendicular to R?

Answer: If the scalar product of M and R is zero, they are perpendicular to each other. Because in scalar product, the intensities of the vectors and the cosine of the angle between them are multiplied. If the angle is 900, the cosine is zero (see: 1.10). In this case, it cannot be reduced to a screw.

 

2. Force Systems

 

We said that if the resultant force and moment are not in the same direction, they will not form a screw.

Figure 2.41

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In summary (Screw Reduction Status is Detected in 2 Steps.)

 

2. Force Systems

 

However, there is a possibility that it will be reduced to a screw. This check is done in Step 2:

 

 

 

 

Calculation of vertical components :

 

Figure 2.42

(2.14)

(2.15)

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Example 2.4 (2006 1st Midterm exam. industrial engineering)

a-) Reduce the force and moment system shown in the figure to point O. (M1 Moment is in the END plane. F1 force is applied from point K in the middle of the CBND plane in the direction of the plane normal.)

b-) Check if the reduced system constitutes a screw,

c-) If it does not constitute a screw, check if it can be reduced to a screw.

d-) If it can be reduced to a screw, reduce it.

2. Force Systems

Solution:

a-)

First, we must find the vector expressions of all forces and moments in order:

 

 

 

 

 

 

 

e-) Calculate the moments of force F2 about the OB and OG axes.

 

 

 

 

 

 

Figure 2.43

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2. Force Systems

 

 

 

 

 

 

 

 

(Since it is in the -y direction):

 

According to the Right Hand Rule, the moment M1 is in the direction of the internal normal of the shaded area. The unit vector in this direction is:

 

 

 

 

 

 

 

 

Figure 2.44

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(Resultant force):

2. Force Systems

 

 

(Resultant Moment):

 

b-) We will check if the reduced system constitutes a screw:

Therefore, it does not constitute a screw.

 

 

 

Figure 2.45

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d-) Let's reduce it to screw :

 

 

c-) Let's check if it can be reduced to a screw:

 

 

 

 

2. Force Systems

 

 

 

 

 

x

y

z

 

 

 

 

 

O

 

 

 

 

 

 

Figure2.46

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Question 2.3

2. Force Systems

 

 

 

Figure2.47

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Question 2.4 (*)

2. Force Systems

Reduce the system consisting of two forces and two moments applied to the prismatic element in the figure to point O. Check whether the reduced system forms a screw.

F1=20kN, F2 = 10kN, C1=8kNm (ADC), C2=5kNm (AOD)

Figure 2.48

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The system consisting of three forces in the figure,

a-) Reduce to point A.

b-) Check whether the reduced system constitutes a screw.

c-) If it does not constitute a screw, check whether it can be reduced to a screw.

d-) If it can be reduced to a screw, reduce it.

2. Force Systems

Question 2.5 (*)

Some Answers:

 

a-)

 

d-)

 

Figure 2.49

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Blank worksheet

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Question 2.6

 

 

Answer:

 

2. Force Systems

Figure 2.50

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A

A mechanic applies 50 N force to the wrench in his hand as shown in the figure and tightens the bolt. Find the moment of this force relative to point A, vectorically.

2. Force Systems

 

Question 2.7

Figure 2.51

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2. Force Systems

Reduce the forces and moments applied to the structure made of pipes to point D.

 

 

Question 2.8

Answer:

Figure 2.52

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Reduce the system consisting of three forces and two moments acting on the pyramid in the figure to point O. Coordinates are given in meters. Moment C1 is applied to the BCD plane, and moment C2 is applied to the OAD plane.

(F1=F2=50kN, C1=C2=20kNm.)

Question 2.9

 

 

 

Figure 2.53

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2. Force Systems

Show that the system consisting of two forces and two moments applied to the element in the figure can be reduced to a screw at point O and reduce it.

 

 

 

Answer:

Question 2.10

Figure 2.54

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F1=10kN

F2=5kN

M1=15kNm

F1=10kN

F2=5kN

M1=15kNm

Reduce the system consisting of forces F1, F2 and moment M1 to a screw at point O. (M1 moment is in the DBHG plane.)

Question 2.11

 

 

(2015 1st Midterm exam- Mechanical Eng.)

 

Answer:

It can be reduced to a screw :

;

2. Force Systems

Figure 2.55

Reduce the system consisting of forces F1, F2 and moment M1 to a screw at point O. (M1 moment is in the DBHG plane.)