PPT: Stress - Strain Transformation

# PPT: Stress - Strain Transformation | Strength of Materials (SOM) - Mechanical Engineering PDF Download

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Chapter 8
Transformation of Stress and Strain;
Yield and Fracture Criteria
Mechanics of Solids
Page 2

Chapter 8
Transformation of Stress and Strain;
Yield and Fracture Criteria
Mechanics of Solids
Fig. 1: State of stress at a point on different planes
Fig. 2: Representations of stresses acting on an element
Page 3

Chapter 8
Transformation of Stress and Strain;
Yield and Fracture Criteria
Mechanics of Solids
Fig. 1: State of stress at a point on different planes
Fig. 2: Representations of stresses acting on an element
Part A- Transformation of Stress
• Transformation of stresses in 2 dimensional problem can be computed
using Fig.3,
Fig. 3: Derivation of stress transformation on an inclined plane
Page 4

Chapter 8
Transformation of Stress and Strain;
Yield and Fracture Criteria
Mechanics of Solids
Fig. 1: State of stress at a point on different planes
Fig. 2: Representations of stresses acting on an element
Part A- Transformation of Stress
• Transformation of stresses in 2 dimensional problem can be computed
using Fig.3,
Fig. 3: Derivation of stress transformation on an inclined plane

?? ??'
= 0 ?? ??'
???? = ?? ?? ???? cos?? cos?? + ?? ?? ???? sin?? sin?? +?? ????
???? cos?? sin?? + ?? ????
???? sin?? cos?? ?? ??'
= ?? ?? cos
2
?? + ?? ?? sin
2
?? + 2?? ????
sin?? cos?? = ?? ?? 1+cos 2??
2
+ ?? ?? 1-cos 2??
2
+ ?? ????
sin2??
?? ?? '
=
?? ?? +?? ?? 2
+
?? ?? -?? ?? 2
cos2?? + ?? ????
sin2?? • Similarly from

?? ??'
= 0,
?? ??'??'
= -
?? ?? -?? ?? 2
sin2?? + ?? ????
cos2?? • Replacing ?? by ?? + 90° gives the normal stress in the direction of the ??'
axis.
?? ?? '
=
?? ?? +?? ?? 2
-
?? ?? -?? ?? 2
cos2?? - ?? ????
sin2??
Page 5

Chapter 8
Transformation of Stress and Strain;
Yield and Fracture Criteria
Mechanics of Solids
Fig. 1: State of stress at a point on different planes
Fig. 2: Representations of stresses acting on an element
Part A- Transformation of Stress
• Transformation of stresses in 2 dimensional problem can be computed
using Fig.3,
Fig. 3: Derivation of stress transformation on an inclined plane

?? ??'
= 0 ?? ??'
???? = ?? ?? ???? cos?? cos?? + ?? ?? ???? sin?? sin?? +?? ????
???? cos?? sin?? + ?? ????
???? sin?? cos?? ?? ??'
= ?? ?? cos
2
?? + ?? ?? sin
2
?? + 2?? ????
sin?? cos?? = ?? ?? 1+cos 2??
2
+ ?? ?? 1-cos 2??
2
+ ?? ????
sin2??
?? ?? '
=
?? ?? +?? ?? 2
+
?? ?? -?? ?? 2
cos2?? + ?? ????
sin2?? • Similarly from

?? ??'
= 0,
?? ??'??'
= -
?? ?? -?? ?? 2
sin2?? + ?? ????
cos2?? • Replacing ?? by ?? + 90° gives the normal stress in the direction of the ??'
axis.
?? ?? '
=
?? ?? +?? ?? 2
-
?? ?? -?? ?? 2
cos2?? - ?? ????
sin2?? ?? ??'
+ ?? ??'
= ?? ?? + ?? ?? • In plane strain problems, where ?? ?? = ?? ????
= ?? ????
= 0, a normal stress
?? ?? can also develop.
?? ?? = ?? ?? ?? + ?? ?? Principal Stresses in Two-Dimensional Problems
• To find the plane for a maximum or a minimum normal stresses,
?? ?? ??'
?? ?? = -
?? ?? -?? ?? 2
2sin2?? + 2?? ????
cos2?? = 0
tan2?? 1
=
2?? ????
?? ?? -?? ??
```

## Strength of Materials (SOM)

37 videos|39 docs|45 tests

## FAQs on PPT: Stress - Strain Transformation - Strength of Materials (SOM) - Mechanical Engineering

 1. What is stress and strain in mechanical engineering?
Ans. Stress is the measure of force per unit area experienced by a material, while strain is the deformation or change in shape experienced by a material when subjected to stress. In mechanical engineering, stress and strain are important concepts used to analyze the behavior and performance of materials under different loading conditions.
 2. How are stress and strain related to each other?
Ans. Stress and strain are related through the material's elastic modulus, which is a measure of its stiffness. The relationship is expressed by Hooke's Law, which states that stress is directly proportional to strain within the elastic limit of a material. This linear relationship is often represented by the equation stress = elastic modulus × strain.
 3. What is stress-strain transformation in mechanical engineering?
Ans. Stress-strain transformation refers to the process of converting stress and strain values from one coordinate system to another. It is commonly used in mechanical engineering when analyzing material behavior under different loading conditions, such as in three-dimensional stress states. The transformation equations allow engineers to determine the principal stresses and strains, as well as the orientation of the principal stress planes.
 4. How is stress-strain transformation important in structural analysis?
Ans. Stress-strain transformation plays a crucial role in structural analysis as it allows engineers to determine the maximum stresses and strains experienced by a structure under various loading conditions. By transforming the stress and strain values to a coordinate system that aligns with the principal directions of stress, engineers can accurately assess the structural integrity and predict potential failure modes.
 5. What are the common methods used for stress-strain transformation?
Ans. There are several methods used for stress-strain transformation in mechanical engineering, including Mohr's circle, graphical methods, and mathematical equations. Mohr's circle is a graphical technique that visually represents stress and strain transformation, allowing engineers to determine the principal stresses and strains. Graphical methods involve plotting stress and strain components on coordinate axes, while mathematical equations involve matrix operations to transform the stress and strain values.

## Strength of Materials (SOM)

37 videos|39 docs|45 tests

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