10.2 TIMOS HENKO BEAM THEORY

In previous section we discussed the gyroscopic effect on a disc alone in a slender flexible beam. The cross-sectional dimensions of the beam were considered to be small in comparison with the length. In this case the rotation of a beam element is equal to slope of the elastic line of the shaft. When we have to investigate higher frequency modes an infinitesimal element would have appreciable amount of rotary inertia . When the beam is thick in that case the cross-sectional dimensions of the beam are considered to be comparable in comparison with the length and the shear effect becomes predominant. The Euler-Bernoulli beam theory is based on the assumption that plane cross-sections remain plane and perpendicular to the longitudinal axis after bending. This assumption implies that all shear strains are zero. When the normality assumption is not used, i.e., plane sections remain plane but not necessarily to the longitudinal (neutral) axis after deformation, the transverse shear strain is not zero. Therefore, the rotation of a transverse normal plane about x-axis is not equal to -dv/dz. Beam theory based on these relaxed assumptions is called a shear deformation beam theory, most commonly known as the Timoshenko beam theory.

Consider a short stubby beam as shown in Figure 10.6(a) before deformation. Let L be the length of the shaft, A be the cross sectional area, E be the Young's modulus, and ρ be the mass density of the shaft material. Assume that the deformation of the beam is purely due to the shear and that a vertical element before deformation remains vertical after deformation and moves by distance v_s (subscript s to represents the pure shear) in the transverse y direction as shown in Figure 10.6(b). For the pure shear case also there will be no coupling in deformations in the two transverse directions (i.e., in the x and y directions).

Since there is no coupling of motion in two transverse directions and on neglecting displacement in the axial direction, the displacement field is given by

Line elements tangential to the elastic line of the beam undergo a rotation β(z,t) due to the shear as shown in Figure 10.6(c). Engineering strains from displacement fields of equation (10.15) are

and

where prime (') represents derivative with respect to z .