Internal Forces in Beams
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.
Concept of Shear Force and Bending moment in beams:
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.
Sign Convention for Shear Force:
The usual sign conventions to be followed for the shear forces have been illustrated in figures 2 and
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.
Concept of Shear Force and Bending moment in beams:
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.
Sign Convention for Shear Force:
The usual sign conventions to be followed for the shear forces have been illustrated in figures 2 and