Viscoelastic beams

Until now, we have considered only elastic beams. To generalize the elastic results to the viscoelastic case is relatively easy. Actually, the correspondence principle (5) indicates that if E tends to E then G approaches G, where the asterisk indicates a complex magnitude. Then, according to Eqs. (17.75) and (17.78), we can write 

In the glassy zone, where the loss is small, Eq. (17.81a) can be simplified to 

The same expression is obtained for the relationship between E and These results lead to tan5 = tanfi in other words, tan 5 is much less sensitive than the moduli to the corrections for shear stresses. The corrections in the moduli depend on the d/i ratio. For dll = 0., the correction is about 3%, of the order of the experimental error, whereas for d// = 0.2, the 

When a beam is loaded at its midpoint, the sections tend to bend. Symmetry considerations indicate that the central section must remain plane. However, since a discontinuous change in the sections adjacent to the central one is not possible, a progressive bending of the cross sections starting from the central one is produced, so that only at a certain distance from the center are the values predicted by theory expected. In other words, in the central section where the load is apphed, the shear forces are smaller than those predicted by theory. Consequently, the deflection will also be smaller. The distribution of stresses in a beam under the action of a concentrated load is an important problem addressed by several authors (see Ref. 6), who concluded that the modulus calculated by means of Eq. (17.82) is overestimated by about 25-30%. 

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The analysis of the stresses and strains in beams and thin rods is a subject of great interest with many practical applications in the study of the strength of materials. The geometry associated with problems of this type determines the specific type of solution. There are cases where small strains are accompanied by large displacements, flexion and torsion in relatively simple structures being the most relevant examples. Problems of this type were solved for the elastic case by Saint Venant in the nineteenth century. The flexion of viscoelastic beams and the torsion of viscoelastic rods are studied in this chapter. 

Equations (17.20) are Laplace transforms of the equations of viscoelastic beams and can be considered a direct consequence of the elastic-viscoelastic correspondence principle. The second, third, and fourth derivatives of the deflection, respectively, determine the forces moment, the shear stresses, and the external forces per unit length. The sign on the right-hand side of Eqs. (17.20) depends on the sense in which the direction of the strain is taken. 

It should be noted that the solution of any viscoelastic beam problem requires knowledge of the boundary conditions at the ends of the rod. These conditions depend on whether the bar is supported, articulated, clamped, free, etc., and for this reason the boundary conditions play a crucial role in the solution of the problem. 

According to the correspondence principle, the equation describing a viscoelastic beam under transversal and longitudinal effects is given by... 

Flexion and Torsion of Viscoelastic Beams and Rods Eq. (17.205) adopts the form... 

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