Section III.1

### Section III.1 Introduction

In previous discussion on pure bending of beams the transverse shear forces and corresponding shear stresses were absent. That absence simplified the flexure problem as the applied loads (i.e., bending moments) only produced flexural normal stresses. In this chapter our focus is on a more general loading condition - one that involves both bending moment and transverse shear force. The existence of the latter indicates that we no longer have pure bending problem.

Recall from mechanics of materials that in beam bending transverse shear forces and bending moments are related to each other. This relationship was displayed in drawing the shear and moment diagrams for various beams and loading conditions.

We first are going to revisit the problems that we encountered in mechanics of materials course. In order to do that we must recall problem restrictions and the method of analysis that was used in the solution of those problems.

Restrictions:

1. Shear stress at every point in the beam must be less than the elastic limit of the material in shear.
2. Normal stress at every point in the beam must be less than the elastic limit of the material in tension and in compression.
3. Beam's cross section must contain at least one axis of symmetry.
4. The applied transverse (or lateral) force(s) at every point on the beam must pass through the elastic axis of the beam. Recall that elastic axis is a line connecting cross-sectional shear centers of the beam. Since shear center always falls on the cross-sectional axis of symmetry, to assure the previous statement is satisfied, at every point the transverse force is applied along the cross-sectional axis of symmetry.
5. The length of the beam must be much longer than its cross sectional dimensions.
6. The beam's cross section must be uniform along its length.

To Section III.2