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Solved Example-1

Here are some examples of using the cut section method to find internal beam forces and moments at specific points:

some typical example

Types of Loads

1- Concentrated load assumed to act at a point and immediately introduce an oversimplification since all practical loading system must be applied over a finite area.

2- Distributed load are assumed to act over part, or all, of the beam and in most cases are assumed to be equally or uniformly distributed.
a- Uniformly distributed load.

a- Uniformly varying load.

Types of Beams

1- Cantilever beam: fixed or built-in at one end while it's other end is free.

2- Freely or simply supported beam: the ends of a beam are made to freely rest on supports.

3- Built-in or fixed beam: the beam is fixed at both ends.

4- Continuous beam: a beam which is provided with more than two supports.
5- Overhanging beam: a beam which has part of the loaded beam extends outside the supports.

Statically Determinate Beams

Cantilever, simply supported, overhanging beams are statically determinate beams as the reactions of these beams at their supports can be determined by the use of equations of static equilibrium and the reactions are independent of the deformation of the beam. There are two unknowns only.

Statically Indeterminate Beams

Fixed and continuous beams are statically indeterminate beams as the reactions at supports cannot be determined by the use of equations of static equilibrium. There are more than two unknown

Concept of Shear Force and Bending moment in beams:

Bending Moment:

Let us again consider the beam which is simply supported at the two prints, carrying loads P1, P2 and P3 and having the reactions R1 and R2 at the supports Fig 4. Now, let us imagine that the beam is cut into two potions at the x-section AA. In a similar manner, as done for the case of shear force, if we say that the resultant moment about the section AA of all the loads and reactions to the left of the x-section at AA is M in C.W direction, then moment of forces to the right of x-section AA must be ?M' in C.C.W. Then ?M' is called as the Bending moment and is abbreviated as B.M. Now one can define the bending moment to be simply as the algebraic sum of the moments about an x-section of all the forces acting on either side of the section
Shear Force
When the beam is loaded in some arbitrarily manner, the internal forces and moments are developed and the terms shear force and bending moments come into pictures which are helpful to analyze the beams further. Let us define these terms Now let us consider the beam as shown in fig 1(a) which is supporting the loads P1, P2, P3 and is simply supported at two points creating the reactions R1 and R2 respectively. Now let us assume that the beam is to divided into or imagined to be cut into two portions at a section AA. Now let us assume that the resultant of loads and reactions to the left of AA is ?F' vertically upwards, and since the entire beam is to remain in equilibrium, thus the resultant of forces to the right of AA must also be F, acting downwards. This forces ?F' is as a shear force. The shearing force at any x-section of a beam represents the tendency for the portion of the beam to one side of the section to slide or shear laterally relative to the other portion.Therefore, now we are in a position to define the shear force ?F' to as follows: At any x-section of a beam, the shear force ?F' is the algebraic sum of all the lateral components of the forces acting on either side of the x-section

Defination

A bending moment exists in a structural element when a moment is applied to the element so that the element bends. Moments and torques are measured as a force multiplied by a distance so they have as unit newton-meters (N·m) , or foot-pounds force (ft·lbf). The concept of bending moment is very important in engineering (particularly in civil and mechanical engineering) and physics.
Tensile stresses and compressive stresses increase proportionally with bending moment, but are also dependent on the second moment of area of the cross-section of the structural element. Failure in bending will occur when the bending moment is sufficient to induce tensile stresses greater than the yield stress of the material throughout the entire cross-section. It is possible that failure of a structural element in shear may occur before failure in bending, however the mechanics of failure in shear and in bending are different.
The bending moment at a section through a structural element may be defined as "the sum of the moments about that section of all external forces acting to one side of that section". The forces and moments on either side of the section must be equal in order to counteract each other and maintain a state of equilibrium so the same bending moment will result from summing the moments, regardless of which side of the section is selected.

Sign Convention

The sign convention used for shear force diagrams and bending moments is only important in that it should be used consistently throughout a project. The sign convention used on this page is as below.

BENDING MOMENT

For the bending moment, following sign conventions may be adopted as indicated in Fig 5 and Fig 6.

Some times, the terms ?Sagging' and Hogging are generally used for the positive and negative bending moments respectively.

SHEAR FORCE

The usual sign conventions to be followed for the shear forces have been illustrated in figures 2 and 3.