Beam deformation angles

The last relation was obtained using Eq. (8.99) for the deformation angle 0. By writing Eq. (8,103) in terms of the Kuhn length, it becomes much more general and applies to beams with cross-sections that are not rectangular (such as the bending of a cylindrical rod). 

Shear deformation occurs in structural elements such as those subjected to torsional loads and in short beams subjected to transverse loads. Shear S-S data can be generated by twisting (applying torque) to a specimen at a specified rate while measuring the angle of twist between the ends of the specimen and the torque load exerted by the specimen on the testing machine (ASTM D 732). Maximum shear stress at the surface of the specimen can be computed from the measured torque that is the maximum shear strain from the measured angle of twist. 

On the other hand, the displacement is accompanied by a rotation of the local surface tangent by a bending angle il> (Fig. 3.7c). Trivially, tan( ) = dw/dx. The calculation of the actual deformation profile w(x) is a common problem in elasticity, known as the bending of a beam without a distributed load. Mathematically, for weak deformation, which is assumed as usual, w(x) has to satisfy the differential equation ... 

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