On the Accuracy of Timoshenko's Beam Theory

The deflection and rotation which appear in Timoshenko's beam theory may be defined either (a) in terms of the deflection and rotation of the centroidal element of a cross-section or (b) in terms of average values over the cross-section. By consideration of an example for which a theoretically exact solution is available it is shown that the Timoshenko equations may be much more accurate if the latter definition is adopted. The example considered is the vibration of a simply-supported beam of narrow rectangular cross-section. The accuracy of the Timoshenko theory depends on the slenderness ratio of the beam, but even when the depth of the beam is equal to the length the Timoshenko theory differs from the exact solution by less than 3.5% provided definition (b) is adopted. Far larger errors occur if definition (a) is used, the most serious error being associated not with shear deformation but with the moment-curvature relation.