INTRODUCTION:
- When a bar carries lateral forces, two important types of loading action are set up at any section: these are a bending moment and a shearing force.
- Consider first the simple case of a beam which is fixed rigidly at one end B and is quite free at its remote end D, Fig1 ; such a beam is called a cantilever, a familiar example of which is a fishing rod held at one end. Imagine that the cantilever is horizontal, with one end B embedded in a wall, and that a lateral force W is applied at the remote end D.
- Suppose the cantilever is dwided into two lengths by an imaginary section C; the lengths BC and CD must individually be in a state of statical equilibrium.
- If we neglect the mass of the cantilever itself, the loading actions over the section C of CD balance the actions of the force Wat C.
- The length CD of the cantilever is in equilibrium if we apply an upwards vertical force F and an anti-clockwise couple A4 at C;
- F is equal in magnitude to W, and M is equal to W(L - z), where z is measured from B.
- The force F at Cis called a shearing force, and the couple M is a bending moment.
Fig 1 Fig 2
- But at the imaginary section C of the cantilever, the actions F and M on CD are provided by the length BC of the cantilever. In fact, equal and opposite actions F and M are applied by CD to BC.
- For the length BC, the actions at Care a downwards shearing force F, and a clockwise couple M.When the cantilever carries external loads which are not applied normally to the axis of the beam, Figure 7.2, axial forces are set up in the beam. If W is inclined at an angle 8 to the axis of the beam, Figure 7.2, the axial thrust in the beam at any section is
- The bending moment and shearing force at a section a distance z from the built-in end are