PPT: Stress - Strain Transformation Notes | EduRev

Strength of Materials (SOM)

Mechanical Engineering : PPT: Stress - Strain Transformation Notes | EduRev

 Page 1


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?? ????
?? ?? -?? ??
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