Beams are structural members designed to support loadings applied perpendicular to their longitudinal axes. Because of these loadings, beams develop an internal shear force and bending moment that, in general, vary from point to point along the axis of the beam. Some beams may also be subjected to an internal axial force; however, the effects of this force are often neglected in the design, since the axial stress is generally much smaller than the stresses developed by shear and bending. A beam that is chosen to resist both shear and bending stresses is said to be designed on the basis of strength.
The stress analysis of a beam generally neglects the effects caused by external distributed loadings and concentrated forces applied to the beam. These loadings will create additional stresses in the beam directly under the load. Notably, a compressive stress will be developed, in addition to the bending stresses and shear stresses discussed previously.
When an engineer is faced with the problem of design using a specific material, it becomes important to place an upper limit on the state of stress that defines the material’s failure. If the material is ductile, failure is usually specified by the initiation of yielding, whereas if the material is brittle, it is specified by fracture. These modes of fracture are readily defined if the member is subjected to a uniaxial state of stress, however, if the member is subjected to biaxial or triaxial stress, the criterion for failure becomes more difficult to establish.
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