**Journal of Theoretical**

and Applied Mechanics

and Applied Mechanics

**36**, 1, pp. 121-144, Warsaw 1998

### The effects of constrained cross-sectional warping on the bending of beams

The equations of motion for a bent beam of compact cross-section, are presented in this work. They are derived by using of the general method, applicable to one-dimensional models of continuum. The warping constraint of cross-sections, caused by shearing, has been taken into account. An additional parameter characterizing the form of a cross-section warping function is introduced. Two dimensionless shearing coefficients appear in the given equations. One of them characterizes the constrained shearing, and the other one the free shearing. It has been shown that these coefficients do not depend upon the form of the warping function. In a particular case, if the constraint cross-sectional warping does not appear, then the equation of motion and the constitutive relations are the same as in Timoshenko's theory.

A series of important examples illustrating the bending theory of beams is presented in the paper. The form of the warping function and the warping constraint of an arbitrary, compact cross-section has been taken into account in these examples. In a particular case, an equation defining the parameter magnitudes of the thickness shear mode, for a simply supported beam of an arbitrary cross-section, is given. A critical analysis the works of Bickford (1982), Ewing (1990), Leung (1990) and Levinson (1981) has been made.

A series of important examples illustrating the bending theory of beams is presented in the paper. The form of the warping function and the warping constraint of an arbitrary, compact cross-section has been taken into account in these examples. In a particular case, an equation defining the parameter magnitudes of the thickness shear mode, for a simply supported beam of an arbitrary cross-section, is given. A critical analysis the works of Bickford (1982), Ewing (1990), Leung (1990) and Levinson (1981) has been made.

*Keywords*: beams; shear effect; vibration