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?? ???? ?? ?? -?? ??Read More

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