Strength of Materials - Lecture Notes / Mehmet Zor

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Separation Principle

and the Concept of Stress

1.2

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• The word stress is used in our daily lives to mean the anxiety and anxiety caused by an event or action.

• After a certain limit, damage such as cracking, rupture, flowing (permanent deformation) may occur in solid objects that experience stress as a result of external loads . This is closely related to the material type of the solid body. The responses of two objects with the same geometry but made of different materials to the same external load will be different. Even if we break a rope by pulling, we cannot break a steel wire of the same thickness.

• Stress in humans occurs due to the pressure of external events. In other words, stress is the effect of external loads (events, pressures) on us. Otherwise, stress is not an external load (pressure). Stress is not applied, it is created.

• If our stress exceeds a certain limit, it causes damage such as shock and fainting. This limit varies from person to person. People who are strong-willed and patient have a high stress limit.

• Similarly, for solid objects, physical loads applied from outside cause stress inside the object.

1.2.1 Stress in humans - Stress in objects

1.2 Separation Principle and Concept of Stress

• Our most important goal in strength is to calculate stresses for different situations. For this, first of all, the separation principle must be understood very well , if this is not understood, it will not be possible to understand the logic of strength.

Figure 1.2.1

Figure 1.2.2

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« If a system subject to the influence of external forces is in equilubrium, each of its parts, which we have separated on an imaginary basis, is also in eqilibrium separately. » This is called the separation principle.

In the free body diagram of each part that we have imaginarily separated, in addition to external forces, internal forces and internal moments are also affected by the separation surface. Each part is in static eqilibrium under the influence of internal and external forces.

stresses

1.2.2 Separation Principle:

Left piece in eqilibrium:

The Whole System is in eqilibrium:

• Internal forces and internal moments ( F _int. , M _int. ): These are the reactions that occur on the separation surface. It is the response of the system to external forces in that part. They are calculated from the static equilibrium of the left or right part of the cut.

stresses

• In statics, the method of sections of trusses systems is actually an application of the separation principle. The forces in the cut bars are internal forces.

1.2 Separation Principle and Concept of Stress

The right piece is in eqilibrium:

Figure 1.2.3.a

Figure 1.2.3.b

Figure 1.2.3.c

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• Stress can be defined as « the effect of internal forces and internal moments per unit area ».

• Understanding the concept of stress is a very important basis for understanding strength issues.

1.2.3. Concept of Stress:

• Unit of Stress is N/mm ² , that is, MegaPascal ( MPa ) . Accordingly, stress can be defined in the simplest sense as the internal force per unit area .

• Stress may vary from point to point. In order to calculate the stress at a point, the internal force and internal moments in the section must first be calculated from static equilibrium with the help of the separation principle.
• We can divide the stress into 2 components: normal and shear stress.
• Stress calculation formulas vary depending on the type of loading (tension-compression, torsion, bending, etc.) and each type of loading constitutes a different subject of strength.

I

F _int.

I

M_int.

S_a

σ _a

τ _a

a

b

S _b

S

Resultant stress

normal stress

n

Shear stress

//n

Plane Normal

S: f ( F _int. , M _int. , geometry)

II section lower part

1.2 Separation Principle and Concept of Stress

Figure 1.2.4.a

Figure 1.2.4.b

Figure 1.2.4.c

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Let's keep in mind : Stress calculation formulas always include internal force or internal moment and values depending on the section geometry. Material properties are (usually) not included in stress formulas. This means that stresses do not depend on the type of material. This is actually an important tip of STRENGTH OF MATERIALS.

1.2 Separation Principle and Concept of Stress

Normal Stresses

We think that in the body below, no internal moment occurs on the separation surface (M_int. = 0) and only the internal force (F_int.) occurs and is distributed homogeneously. In this case, we can define stress as internal force per unit.

Figure 1.2.5.a

Figure 1.2.5.b

Figure 1.2.5.c

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1.2.4. Separation Principle - Example

I

I

I

I

Examine the stresses occurring in different parts of a standing person due to his body weight. Please note that imaginary cuts are made for the regions, and only internal force occurs in each cut.

II

II

1.2 Separation Principle and Concept of Stress

A _body

Figure 1.2.6.a

Figure 1.2.6.b

Figure 1.2.6.c

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• Examining the bottom or top of the cut does not change the result. This situation is proven in the figures below.

II section Upper part

II section Lower part

1.2 Separation Principle and Concept of Stress

A _neck

• In the figure below, separate examinations were made for the lower and upper parts of section I-I.
• Notice that there is the same internal force F _int.-1 in the neck region for both sections and therefore the same σ_neck stress value.
• This is a general rule and inspection of any part of the cut (top or bottom; right or left) is sufficient.

W _all / 2

W _all / 2

Figure 1.2.6.d

Figure 1.2.6.e

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1.2.5 Stress Types

1.2.5.1 Normal Stress: The stress parallel to the plane normal, in other words, perpendicular to the plane, is called normal stress; It is denoted by σ .

Imaginary

separation surface

Upside

Under Part

(1.1)

1.2 Separation Principle and Concept of Stress

(a)

Figure 1.2.7

(b)

(c)

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1.2.5.2 Shear Stress : The stress that is perpendicular to the plane normal, in other words parallel to the plane, is called normal stress; It is denoted by τ .

right part

In this example, since we neglect the difference in perpendicular distance between F_int. and F the internal moment is zero.

(1.2)

1.2 Separation Principle and Concept of Stress

Figure 1.2.8

(a)

(b)

(c)

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1.2.5.3 Stress indices

σ or τ We define the stresses in more detail and accurately by adding 2 indices:

x

z

y

1st subscript : i : It shows the normal of the plane where the stress is located.

2nd subscript : j :Indicates the axis of stress.

• Try to understand that the stress indices in the cubic element on the side are put in accordance with this definition. Identify the stress with a question mark along with its subscript.

plane normal

Stress

axis

?

• σ normal stresses are always in the line of the plane normal and both indices are the same .

1.2 Separation Principle and Concept of Stress

Figure 1.2.9

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Let's show the shear stresses on each surface of a cubic differential element, having a.b.t dimensions, at a point O in the xy plane.( t: thickness ) :…>>>

Similarly, the same results are found for other planes:

This element must be in static eqilibrium. Equilibrium equations should be written for forces.

1.2.5.4 Specific information on shear stresses

This can be proven as follows:

a-) Shear stresses with the same indices but different subscript orders have equal intensity.

Force = stress x area

1.2 Separation Principle and Concept of Stress

From these 3 equations

is available.

If we express the stresses with their indices :

plane normal

stress direction

τ _{i j} = τ _{j i}

Figure 1.2.11

Figure 1.2.10

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It should not be forgotten that if the material of the rope changes, the load W that can be carried will also change.

1.2.6 Damage Stress

Let's consider the load W hanging on the rope in the figure. At any time the force in the rope is F_int. = W from static equilibrium .

In the table below, the force (F_break) values ​​in the rope that occur at the instant of breaking are shown according to the cross-sectional area of the rope.We know that as the cross-section grows, that is, the thicker the rope, it will carry more W load.

Question: Is there a value that remains constant according to this table?

(1.3)

1.2 Separation Principle and Concept of Stress

Figure 1.2.12

Tablo 1.2.1

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Tensile Testing Machine

Ductile (formable) materials, the time when permanent deformation (plastic zone) begins should be considered as the instant of damage. Because an element exposed to permanent deformation will lose its functionality. Therefore, the damage stress is:

Note: The yield strengths (yield stresses) of ductile materials in tension and compression are the same. In brittle materials, the compressive strength is greater than the tensile strength (may be 3-4 times).

1.2.7 Damage Stress detection

(1.4)

(1.5)

1.2 Separation Principle and Concept of Stress

Figure 1.2.14

Figure 1.2.13

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1.2.8. Damage Condition for Tension-Compression Situations

Damage condition for ductile materials:

Damage requirement for brittle materials:

(The reason why we included it in absolute value is to cover the damage condition in case of compression. There is no need to take it in absolute value in case of tension.)

It hasn't broken yet.

instant of rupture

Pull !

a little more

Okey brother

I’m pulling

Finally! We broke it off

Why did we break this now???

P=270N

P=270N

P=350N

P=350N

Cross-sectional area: A=10mm ²

for rope

Try to understand the concepts thoroughly by carefully examining the rope breaking example below.

LATER

I

I

Cut I-I

I

I

Normal stress occurs in a material subjected to tensile loading. As the force increases, the tension also increases. If the force is increased significantly, the stress reaches the damage stress and damage occurs in the material. Then, in general terms, the damage condition is that the current stress is equal to or greater than the damage stress.

(Damage condition).

1.2 Separation Principle and Concept of Stress

Figure 1.2.15.a

Figure 1.2.15.b

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1.2.9 Let's explain the damage conditions with some different symbols.

For ductile materials , the yield stress in tension is

Tensile breaking stress for brittle materials

For ductile materials , the yield stress in compression is

Breaking stress in compression for brittle materials

In other words, if at least one of the two inequalities above is met, at that point, yielding occurs in ductile materials and breaking occurs in brittle materials.

or

whereas,

The breaking stresses of brittle materials in compression are significantly greater than those in tension.

Tensile and compressive yield stresses of ductile materials are equal in intensity.

This is because the internal structure of brittle materials is porous. The pores in the material close in case of compression, the surface area expands and the material can withstand more compressive load.

Attention: Since these symbols are used in some sources, it is useful to learn them.

At this point;

1.2 Separation Principle and Concept of Stress

Figure 1.2.16

Figure 1.2.17

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1.2.10 We can now explain some of the things we know from experience in a more technical way:

Stress occurring in case of compression or tension:

We learned that it was.

Because the contact area of the sharp corners of a table or object is very small, like the tip of a needle, even the slightest blow to the sharp corners instantly causes exceeding the damage stress in that part of our body and serious injuries occur.

If the tips of our fingers were not rounded, we would probably feel great pain when touching something and we would not be able to use our hands fully. Similarly, see the other roundings on your body, realize that you are a perfect design even from this aspect alone .

For this reason, when designing a product, we increase the contact area by giving fillet (rounding) to sharp corners and reduce the instantaneous stress.

Protective apparatus for babies..

1.2 Separation Principle and Concept of Stress

Figure 1.2.18

(a)

(b)

Figure 1.2.19

Figure 1.2.20

(c)

Strength of Materials - Lecture Notes / Mehmet Zor

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1.2.11 Damage Condition in Case of Shearing

For ductile materials:

For brittle materilas:

(Whether the sign of the shear stress is + or – is not important in terms of strength. It is considered as + here.)

(The reason for this equality will be better understood from the Mohr circle topic, which will be seen.later.)

Damage condition:

upper part of cut E-E

We think that there is a tensile load on the plates.

Forces acting on the bolt

Internal force(F_int.) in cross-section of the bolt

Stress occurring in cross-section of the bolt

(1.6)

(1.7)

1.2 Separation Principle and Concept of Stress

An example : We check the damage of the steel bolt connecting two plates with the following steps.

Figure 1.2.21

(a)

(b)

(c)

(d)

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1.2.12 Factor of safety and allowable stress

• It is desired that a mechanism, structure or machine have a certain safety factor.
• In other words, the system can be designed to carry 2 or 3 times the load it can carry.
• It is desirable that this coefficient be higher, especially in mechanisms such as elevators where life safety is important.
• In this case, strength calculations are made according to the allowable stress, not the damage (yield or breaking) stress.
• Dimensions are calculated or material selection is made so that the maximum stress in the structure does not exceed the allowable stress.

Safety condition:

or

Reasons for Using the Safety Coefficient:

• Uncertainties and variations in material properties
• Uncertainties and variations in loadings
• Errors (mistakes) and uncertainties in analyzes
• Repetitive loading situations
• Damage types
• Degrading effects on the material and repair requirements
• Life and property safety/security
• Functionality of the machine

(1.8)

(1.9a,b)

1.2 Separation Principle and Concept of Stress

Figure 1.2.22

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1-) In an object or system in equilibrium, unknown forces are first found through static calculations.

2-) The maximum stress on the object is calculated.

(Calculation method and formulas vary depending on the loading method and sometimes geometry.)

1.2.13 General Road Map in Strength Calculations

3-) The maximum stress is equal to the yield stress in ductile materials and the breaking stress in brittle materials. From this equality,

a-) minimum size calculation (if force + material is certain) or

b-) material selection (if force + dimensions are certain) or

c-) Strength (strength) control is performed (if force + dimensions + material type is known).

• This is the general roadmap. In some problems, the maximum force that can be applied to the object may be asked.
• Or strains as well as stresses can be calculated. The factor of safety can be given.
• With the sample problems to be seen in the future, these will become more clear in your mind and the logic of resilience will become clearer.

• Important Tip : For all strength topics, you will see the terms force or moment in stress or strain formulas. Note that these terms always refer to the internal force or internal moment in the section under consideration. The internal force or internal moments must be found in advance by the separation principle so that the stresses can be calculated.

1.2 Separation Principle and Concept of Stress

Figure 1.2.23