Introduction: A structural member which is acted upon by a system of external loads at right angle to its axis is known as beams. We see that whenever a horizontal beam is loaded with a vertical loads sometimes, it bends due to the action of the loads. The amount with which a beam bends, depend upon the amount and type of the loads, length of the beam, elasticity of the beam and type of the beam. The scientific way of studying the deflection or any other effects is to draw and analyse the shear force or bending moment diagram of a beam.
Types of beams
The types of beams are classified as under:
- Cantilever beam
- Simply supported beam
- Overhanging beam
- Rigidly fixed or built in beam
- Continuous beam
It is difficult to accurately estimate the loads coming on a structure. They can be caused by gravity, wind pressure, hydrostatic pressure, acceleration, temperature changes, shrinkage, friction, earthquake, impact and vibration. A beam maybe subjected to either or in combination, Loads are usually modelled as:
- concentrated or point load
- Uniformly distributed load
- Uniformly varying load
- Concentrated couple
Loads are generally classified in two groups
Dead loads: are weight of the structural system and are computed on the basis of weight density and dimensions of the structures. In a building this includes the self wight of all beams, columns, trusses, walls etc and other service equipment supported by the structure.
Live/imposed loads: These are mobile loads to be carried by the structure and because of their nature are more difficult to determine precisely.
The shear force at the cross section of a beam, is defined as the unbalanced vertical forces to the right or left of the section.
The bending moment at the cross section of a beam, is defined as the algebraic sum of the moment of the forces , to the right or left section.
Sign conventions for shear force and bending moment
We find different sign conventions in different books, regarding shear force and bending moment at a section .we are going to use the following sign convention which is widely and internationally recognized.
- Shear force: we know that as the shear force is the unbalanced vertical forces, therefore it tends to slide one portion of the beam, upward or downwards with respect to the other.we take the shear force at a section as positive, when the left hand portion tends to slide upward or the right hand portion tends to slide downward. Similarly , we take shear force at a section as negative, when the left hand portion tends to slide downward or the right hand portion tends to slide upward.
- Bending moment: we take the bending moment at a section as positive, if it tends to bend the beam at that point to a curvature having concavity at the top. On the other hand , we take the bending moment at a section as negative, if it tends to bend the beam at that point to a curvature having a convexity at the top. We often call the positive bending moment as sagging moment and negative bending moment as hogging moment.
Another way of assigning the sign convention to the bending moment is by the direction in which it acts at a section . we take the bending moment at a section as positive, when it’s acting in a clockwise direction to the left or in anticlockwise direction to the right.
On the other hand , we take the bending moment at a section as negative, when it’s acting in anticlockwise direction to the left or in a clockwise direction to the right.
Note: while calculating bending moment or shear force , at a section the beam is assumed to be weightless.
Relation between loading, shear force and bending moment
The following relations between
loading,shear force and bending moment at a point or between any two sections of a beam are important from the subject point of view:
- If there is a point load at a section on the beam, then the shear force suddenly changes ( the shear force line is vertical). But the bending moment remains the same.
- If there is no load between two points, then the shear force does not change ( shear force line is horizontal) but the bending moment changes linearly ( bending moment line is inclined straight line)
- If there is a uniformly distributed load between two points, then the shear force changes linearly ( shear force line is an inclined straight line). But the bending moment changes according to the parabolic law. (Bending moment line will be a parabola)
- If there is a uniformly varying loads between two points then the shear force changes according to the parabolic law( shear force line will be parabola). But the bending moment changes according to the cubic law.