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By O A Bauchau

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Additional info for Structural analysis: with applications to aerospace structures

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49c) These important results show that knowledge of the stress components σ1 , σ2 , and τ12 on two orthogonal faces allows computation of the stress components acting on a face with an arbitrary orientation. In other words, the knowledge of the stress components on two orthogonal faces fully defines the state of stress at a point. 5 Special states of stress Two plane stress states are of particular interest. One is called the hydrostatic stress state and the other is called the pure shear state.

The relative elongations are also called direct strains or axial strains, whereas the angular distortions are called shearing strains or simply shear strains. It is important to note that the strain-displacement relationships, eqs. 71) are obtained under the small displacement assumption defined in eq. 61). 67) should be used instead. , | 1| 1, | 2 | 1, | 3 | 1, |γ23 | 1, |γ13 | 1, |γ12 | 1.

The solutions of eq. 14) are called the principal stresses. Since this is a cubic equation, three solutions exist, denoted σp1 , σp2 , and σp3 . For each of these three solutions, the matrix of the system of equations defined by eq. 13) has a zero determinant, and a non-trivial solution exists for the directions cosines that now define the direction of a face on which the shear stresses vanish. This direction is called a principal stress direction. Because the equations to be solved are homogeneous, their solution will include an arbitrary constant, which can be determined by enforcing the normality condition for unit vector n ¯ , n21 + n22 + n23 = 1.

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