4.1 INTRODUCTION

In the previous chapter, we have seen how to draw shear force diagrams (SFDs) and bending moment diagrams (BMDs). The question that arises is that, how these bending moment (BM) and shear force (SF) should be handled in design and what type of stresses develops due to these internal forces? In this chapter, we will discuss about the quantifi cation of stress developed due to BM. The stress that develops due to bending is normal stress generally referred as fl exural stress or bending stress.

4.2 ASSUMPTIONS IN THEORY OF BENDING

• 1. Material of the beam is homogeneous, elastic, and isotropic.

• A material is said to be elastic, if the force–displacement relationship follows the same path during the loading as well as during the unloading. In homogeneous materials, the elastic properties such as modulus of elasticity and Poisson's ratio is same everywhere. A material is said to be isotropic, if the elastic properties are same in all directions.

• 2. Material of the beam obeys Hooke's law, that is, force–displacement relationship is linear.

• 3. Plane sections normal to the longitudinal remains plane even after bending.

• This is the most important assumption; it means that the strain variation is linear across the depth of the beam cross-section.

• In Figure 4.3, cantilever beam is subjected to moment M. AB is a plane section, before the application of the couple. After the application of the couple, the plane section AB has rotated about C, taken position A'B. As the A'B' is also a straight line, the displacement of any point on AB is proportional to its distance from C. Hence, the strain at any point on the cross-section AB is proportional to its distance from C.

• 4. The beam is subjected to pure bending.

• Pure bending means the beam should be subjected to constant BM and no SF and axial force act on it, for example,

• In the beam shown in Figure 4.4 (a), the portion BC of the beam ABCD is subjected to pure bending. In Figure 4.4 (b), the beam AB is subjected to pure bending.