Stress Strain Relationships for Off-Axis Orientation

The orthotropic stress and strain relationships of Equations 8.42 and 8.43 were defined in principal material directions, for which there is no coupling between extension and shear behavior. However, the coordinates natural to the solution of the problem generally will not coincide with the principal directions of orthotropy. For example, consider a simply supported beam manufactured from an angle-ply laminate. The principal material coordinates of each ply of the laminate make angles 0 relative to the axis of the beam. In the beam problem stresses and strains are usually defined in the beam coordinate system (jc,y), which is off-axis relative to the lamina principal axes (L, T). 

Recall that the Hooke s law for an isotropic material remains the same regardless of the orientation of the stress element being considered. For example, if the state of plane stress is known at a point in the (jc, y) plane (i.e., o, Oy, is known at a point), the state of strain at the point (e, e, %y) can be determined using Equation 8.34. Similarly, if we know the stress Oy, relative to a new set of coordinates (x, y ) rotated relative to the (x, y) axes, the strains ey, can be determined using the 

FIGURE 8.15 Positive rotation of principal material axes (L, T) from arbitrary reference 

The matrices [5] and [Q] appearing in Equations 8.42 and 8.43 are called the reduced compliance and stiffness matrices, while [5 ] and Q in Equations 8.54 and 8.55 are known as the transformed reduced compliance and stillness matrices. In general, the lamina mechanical properties, from which compliance and stiffness can be calculated, are determined experimentally in principal material directions and provided to the designer in a material specification sheet by the manufacturer. Thus, a method is needed for transforming stress-strain relations from off-axis to the principal material coordinate systan. 

To begin the analysis, recall the stress transformation equations from elementary mechanics of materials. In a first course on mechanics of materials the transformation 

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