Question
Download Solution PDFIn generalized three - dimensional state of stress, the number of independent stress components is
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFConcept:
Three-Dimensional State of Stress:
In a three-dimensional stress state, the stress at a point is represented by a stress tensor, which is a 3 × 3 matrix. This matrix includes normal and shear stress components.
The general form of the stress tensor is:
Where:
- σxx, σyy, σzz are the normal stress components.
- τxy, τyz, τzx are the shear stress components.
Due to the symmetry of the stress tensor, shear stress components on mutually perpendicular planes are equal (i.e., τxy = τyx, τyz = τzy, τzx = τxz).
Calculation:
The number of independent stress components in a three-dimensional state of stress is calculated as follows:
- 3 normal stress components: σxx, σyy, σzz
- 3 shear stress components: τxy, τyz, τzx (since τxy = τyx, τyz = τzy, τzx = τxz)
Total number of independent stress components = 3 (normal) + 3 (shear) = 6
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