Data for Q3–Q4 are given below. Solve the problems and choose correct answers
4. Bending Moment and Shear Force Diagram
Theory at a Glance (for IES, GATE, PSU)
4.1 Shear Force and Bending Moment
At first we try to understand what shear force is and what is bending moment?
We will not introduce any other co-ordinate system.
We use general co-ordinate axis as shown in the figure. This system will be followed in shear force and bending moment diagram and in deflection of beam.
Here downward direction will be negative i.e.
negative Y-axis. Therefore downward deflection of the beam will be treated as negative.
We use above Co-ordinate system
Some books fix a co-ordinate axis as shown in the following figure. Here downward direction will be positive i.e. positive Y-axis. Therefore downward deflection of the beam will be treated as positive. As beam is generally deflected in downward directions and this co-ordinate system treats downward deflection is positive deflection.
Some books use above co-ordinate system
Consider a cantilever beam as shown subjected to external load ‘P’. If we imagine this beam to be cut by a section X-X, we see that the applied force tend to displace the left-hand portion of the beam relative to the right hand portion, which is fixed in the wall.
This tendency is resisted by internal forces between the two parts of the beam. At the cut section a resistance shear force (Vx) and a bending moment (Mx) is induced. This resistance shear force and the bending moment at the cut section is shown in the left hand and right hand portion of the cut beam.
Using the three equations of equilibrium
0 , 0 0
x y i
F F and M
¦ ¦ ¦
We find that Vx P and Mx P x.
In this chapter we want to show pictorially the
Chapter-4 Bending Moment and Shear Force Diagram S K Mondal’s variation of shear force and bending moment in a
beam as a function of ‘x' measured from one end of the beam.
Shear Force (V) ŋ equal in magnitude but opposite in direction to the algebraic sum (resultant) of the components in the direction perpendicular to the axis of the beam of all external loads and support reactions acting on either side of the section being considered.
Bending Moment (M) equal in magnitude but opposite in direction to the algebraic sum of the moments about (the centroid of the cross section of the beam) the section of all external loads and support reactions acting on either side of the section being considered.
What are the benefits of drawing shear force and bending moment diagram?
The benefits of drawing a variation of shear force and bending moment in a beam as a function of ‘x' measured from one end of the beam is that it becomes easier to determine the maximum absolute value of shear force and bending moment. The shear force and bending moment diagram gives a clear picture in our mind about the variation of SF and BM throughout the entire section of the beam.
Further, the determination of value of bending moment as a function of ‘x' becomes very important so as to determine the value of deflection of beam subjected to a given loading where we will use the formula,
2
2 x
EId y M
dx .
4.2 Notation and sign convention x Shear force (V)
Positive Shear Force
A shearing force having a downward direction to the right hand side of a section or upwards to the left hand of the section will be taken as ‘positive’. It is the usual sign conventions to be followed for the shear force. In some book followed totally opposite sign convention.
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Chapter-4 Bending Moment and Shear Force Diagram S K Mondal’s
The upward direction shearing force which is on the left hand of the section XX is positive shear force.
The downward direction shearing force which is on the right hand of the section XX is positive shear force.
Negative Shear Force
A shearing force having an upward direction to the right hand side of a section or downwards to the left hand of the section will be taken as ‘negative’.
The downward direction shearing force which is on the left hand of the section XX is negative shear force.
The upward direction shearing force which is on the right hand of the section XX is negative shear force.
x Bending Moment (M) Positive Bending Moment
A bending moment causing concavity upwards will be taken as ‘positive’ and called as sagging bending moment.
Chapter-4 Bending Moment and Shear Force Diagram S K Mondal’s
Sagging
If the bending moment of the left hand of the section XX is clockwise then it is a positive bending moment.
If the bending moment of the right hand of the section XX is anti-clockwise then it is a positive bending moment.
A bending moment causing concavity upwards will be taken as ‘positive’ and called as sagging bending moment.
Negative Bending Moment
Hogging If the bending moment of
the left hand of the section XX is anti-clockwise then it is a positive bending moment.
If the bending moment of the right hand of the section XX is clockwise then it is a positive bending moment.
A bending moment causing convexity upwards will be taken as ‘negative’ and called as hogging bending moment.