Test: Shear Force & Bending Moment - 1 - Civil Engineering (CE) MCQ

The shear force in the span BC will be zero. Hence the options A and B are incorrect. The shear force in the span AB and CD will be of opposite sign. Therefore, the option C is correct.

Let the point of zero shear force occur at section X-Xfrom left support A
∴ Shear force at X-X

∴ For zero shear force we have, 

∴ 

The BM can be found from the area of SFD.
The shear force diagram is

Maximum negative, bending moment,

Maximum positive, bending moment,

For M₁ = M₂

Position of Maximum Bending moment under a particular point load among a train of point loads:



Conclusion: The maximum bending moment under a particular point load among a train of point loads crossing a simply Supported girder occurs at the location when that load is at a point so that the load and the resultant should be equidistant from the mid-span.

The relation between shear force (V) and bending moment (M) is:

dM/dx = V

it means the slope of a bending moment diagram will represent the magnitude of shear force at that section.

Since the bending moment is constant along the length, therefore its derivative i.e. shear force is equal to zero at all sections along the beam.
There is no shear force between the loads and the bending moment is constant for that section along the length and vice-versa.

Additional Information

The relation between shear force (V) and loading rate (w) is:

dV/dx = w

it means a positive slope of the shear force diagram represents an upward loading rate.

The relation between loading rate and shear force can be written as:

Explanation:

The following point should always be kept in mind while drawing SFD and BMD:

1. The rate of change of the shear force at any point on the axis of the beam is equal to the negative of the intensity of the distributed load at that same point.

i.e. w=−dV/dx

Here, negative sign represents that the loading is acting downward.

2. The rate of change of the bending moment at any point on the axis of a beam is equal to the shear force at that same point.

i.e. V=dM/dx

Calculation: -

From (2),

We have,

dM/dx=V=constant

Thus,

The magnitude of slope of moment diagram is also constant.

Bending moment (BM):

• It is a measure of the bending effect that can occur when an external force (or moment) is applied to a structural element.
• Failure can occur due to bending when the tensile stress exerted by a force is equivalent to or greater than the ultimate strength (or yield stress) of the element. However, although the mechanisms are different, a beam may fail due to shear forces before failure in bending.

Factors affecting bending moment:

1) Position and magnitude of the load

2) Type of beam: For a simply supported beam with UDL throughout the span, the maximum bending moment (WL²/8) is more as compared to a fixed beam(WL²/12) with the same loading condition.

3) Span of the beam: 

For example:

• A simply supported beam subjected to a uniformly distributed load will have a maximum bending moment at the center.
• Simply supported beams of two continuous spans subjected to uniformly distributed load would have a maximum sagging moment at the span center and maximum hogging moment at the supports.

Span AC:

Span CB:

Points of zero bending moment

• The points of contra flexure are points of zero bending moment, i.e. where the beam changes its curvature from hogging to sagging.
• In a bending beam, a point of contra flexure is a location where the bending moment is zero (changes its sign).
• In a bending moment diagram, it is the point at which the bending moment curve intersects with the zero lines.

Point of inflexion:

• Point of inflexion is the point where the elastic deflection curve changes its curvature not bending moment diagram.