4. PLANE TRUSSES

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4a

(tvid – 4)

Trusses

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They are carrier systems formed by rod elements connected to each other at their end points.

4- Plane Trusses

Figure 4.1

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There are many application areas.

​

It is used in

• roof systems,
• Bridges,
• Towers,
• and many similar structures.

4- Plane Trusses

Figure 4.2

Figure 4.3

Figure 4.4

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4.1 Main Features of Trusses:

1. The rods are connected at their ends and these end connection points are called "joints".
2. External forces only act on the joints.
3. External connections should only be at the joints.
4. Only single forces occur at the joints.
5. Moment reactions in the connections are neglected.
6. Rod weights are neglected in the solutions.
7. Each rod experiences a force in its own direction (tensile or compressive force).

Systems that do not have at least one of these features are not considered truss systems and fall within the scope of the frame system that will be shown in the next topic.

4- Plane Trusses

Figure 4.5

Figure 4.6

Figure 4.7

Figure 4.8

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4.2 Types of Truss Systems:

4.3 Our aim in this subject within the scope of Statics Course is to calculate the forces acting on each rod or specific rods, while the geometry of the Truss System and external forces are known. Only plane truss systems are included in the scope of the course.

1- Space Truss Systems: They are 3-dimensional systems.

2- Plane Truss Systems: They are 2-dimensional systems.

4- Plane Trusses

Figure 4.9

Figure 4.10

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F

F

F

Force applied externally to the finger

Force from joint to finger

Force from finger to joint

force coming from other parts of the hand to the joint

F

eklem

• Forces are distributed in truss systems as in this logic.
• Each bar carries a tensile or compressive force in the direction of

its own axis.

• The forces come from the joints of the rod, with equal intensity, in opposite directions, and on the line connecting the joints (the axis of the rod). (This means that each bar is a two force member. Two force members will be explained in more detail in the subject of frames.)

4.4 Force Distribution Logic in Truss Systems

4- Plane Trusses

• In Example 4.1, force distribution and force calculation methods will be better understood.

• To better understand the force distribution in truss systems, use your other hand to pull your index finger.

• Try to understand the force distribution in the figure. In the meantime, you should realize that both the finger and the joint are in equilibrium. It can be said that, your finger = a rod in truss system,

your joint = a connection point (a joint) of the truss system.

Figure 4.12

Figure 4.11

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4.5 Rod Forces and Calculation Methods:

We will try to understand this issue with an example.

First we must understand the logic of force distribution in rods and joints.

4- Plane Trusses

Example 4.1) Calculate the force that occurs in each rod in the plane truss system formed by 5-meter rods.

30kN

10kN

Using the finger example on the previous page, let's draw the force distribution between rods AB and AC and joints A, B and C:

Tip 5.1) How do we choose the direction of the forces? When a bar force is first placed, an arbitrary direction (tension or compression) is selected, provided that it is parallel to the bar axis. However, the direction of the same force cannot be selected arbitrarily in the 2nd, 3rd, placements. It is selected depending on the first placement. For example, the FAC force is first applied to the A joint, arbitrarily to the left. This force must necessarily be to the right (action-reaction) at the A end of the AC bar and to the left at the C end of the AC bar so that the bar is balanced. At joint C, it should be towards the right (action-reaction). If the sign of the force is «-» as a result of the calculations, it shows that it is in the opposite direction to the direction we selected. However, in this case, the direction of the force is not reversed, it is used with the «-» sign in the calculations. If it is reversed, its sign must also be changed.

Solution:

F_AC First Direction:

arbitrary

mandatory

Figure 4.13

Figure 4.14

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• If you pay attention, each joint receives a force from each of the rods connected to it.
• In addition, external forces can be applied to the joints. (External forces do not affect the middle regions of the rod.)
• On the other hand, the rods receive a reaction force of equal strength and opposite direction from each of the joints to which they are connected (action-reaction).
• Rod forces are called the internal forces of the system and in fact their sum is zero.
• A rod force is necessarily in the direction of the rod axis.
• Because each rod is a two force member. (Two force member will be explained in topic 5.4.)

4- Plane Trusses

Force distribution in the entire system:

Figure 4.15

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We continue the same example…

First, the forces at the connection points are calculated in the equilibrium of the entire system:

Tip 5.2: In some problems, the desired rod forces can be found without having to calculate the reaction forces in the connections. It is useful to solve many questions to see this situation and get used to it. In the cutting method, if all the connections are on one side of the cut, there is no need to find the reaction forces… The other side of the cut is examined and the rod forces can be found.

In the balance of the entire system, the system is isolated only from external connections. (In this isolation, the rods are not cut, so the rod forces remain as internal forces. For this reason, they are not included in the equilibrium equations of the entire system.)

bulunur.

4- Plane Trusses

T

Now we will calculate the forces acting on the rods and joints. There are 2 methods for this:..>>

Figure 4.16

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4.5.1 ) Method of Joint

4- Plane Trusses

After A, we can move on to joint B. Because FAB is found, there are 2 unknown forces left in B.

Since there are 2 unknown forces left at joint C, we can move on to C.

Figure 4.17

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Method of Section

4- Plane Trusses

4.5.2

I

I

II

Cut I-I

Left section (FBD)

From the eqilibrium of the left section:

We found the same results as in the Joint Method

Since we can apply 3 equations of equilibrium in a cut, there must be at most 3 unknown bar forces so that all of them can be found. For example, if we had examined the II-II cut first, we would not be able to find the forces of all 4 cut bars.

II

Figure 4.18

Figure 4.19

The truss system in the example we examined is in balance under the influence of external forces. Therefore, the left and right parts (sections) of the imaginary cuts we made are also in balance separately. (Separation Principle)

Note: Since we are placing it for the first time, we could have chosen a rod force in the opposite direction. In this case, the magnitude of the calculations would not change, but the sign would be the opposite. For example, if we had chosen FBC from B to C, its value would be -34.64kN.

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4- Plane Trusses

I

I

If we had examined the Right part of the I-I cut, we should have found the same results. Let's prove it:

(Same Results)

II

II

Figure 4.20

Figure 4.21

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4.7 Zero Force Member:

In general; The zero force member makes a 900 angle with the other rods in a joint and no external force acts on that joint in the direction of the rod. Considering the balance of the joint B for the example above:

How is it detected?

Hz. Mevlana

It is the name given to rods on which no force falls, that is, whose tensile or compressive force is zero.

4- Plane Trusses

Rod GB is a zero force member.

4.6 Wrong Cuts:

1. A cut that passes through one or more joints, such as the a-a cut, cannot be made.

a

a

Wrong cut

b

b

wrong cut

c

c

Right cut

2. A cut cannot be made halfway through, such as a b-b cut.

3. The cut (like a c-c cut) should be made from the places between the two joints, corresponding to the middle areas of the rods, and from length to length. The right or left part should be completely separated from each other. So much so that the left or right part can be taken away.

Question: Do you think the d-d cut is a correct cut? Discuss among yourselves.

d

d

Figure 4.22

Figure 4.23

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Example 4.2 : Find the rod forces in the given truss system using the method of joint.

Solution: First of all, the reaction forces in the connections are found in the equilibrium of the entire system.

FBD (Entire System)

4- Plane Trusses

Figure 4.24

Figure 4.25

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4- Plane Trusses

Figure 4.26

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Let's examine joint C for control purposes:

4- Plane Trusses

(for control purpose)

These equations are also satisfied. It supports the accuracy of the results we found above.

Figure 4.26

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Find the rod forces ED and EF by using the a-a cut.

a-a cut

Example 4.3 :

4- Plane Trusses

Solution:

(Other rod forces cannot be found from the a-a cut alone.)

If we examine the equilibrium of joint E:

Figure 4.27

Figure 4.28

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Example 4.4 In the truss system in the figure

a-) Find the forces in the GE, GC and BC bars.

b-) Is there an zero force member in the system? Determine.

Solution:

Let's find the support reactions from the equilibrium of the entire system:

4- Plane Trusses

Let's examine the equilibrium of the left side of the f-f cut :

Figure 4.29

Figure 4.30

Figure 4.31

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Question 4.1 (*): Find the force on the rod CF in the truss system shown in the figure.

4- Plane Trusses

Figure 4.32

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In the Truss Systems Below a-) Calculate the forces in the rods with question marks. b-) Identify the zero force members, if any. (Answers are given next to the questions. The method is free.)

4- Plane Trusses

Answers:

AB= -750N,

BC = -600N,

BD = 250N

Question 4.2(*):

Question 4.3

Answers:

CF= 16kN,

CE = CA= -11.31kN

Figure 4.33

Figure 4.34

Zero force Member: DE

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4- Plane Trusses

Zero force Members : CH, JI, GF, EF

Answer: DE = 146.41kN

Question 4.4

Question 4.5

Answers: GH =-20kN, CD =40kN , DH =-28.28 kN

Figure 4.35

Figure 4.36

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Answer: CD=1.87kN

4- Plane Trusses

Question 4.6

Question 4.7

Figure 4.37

Figure 4.38

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Zero force Members : AF, AB, DE, DC

Answers: CB = 66.6kN ; EC = 83.3kN; EB = -50kN

Question 4.8

Question 4.9

Answers : EF = -4kN ; CF = -2.237kN; FB = -1kN

Figure 4.39

Figure 4.40

4- Plane Trusses