Strength of Materials : Shear Force and Bending Moment Diagrams Notes

By Deepanshu Rastogi|Updated : December 3rd, 2021

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             

 

Shear Force and Bending Moment Diagrams

Beam is one of the most important structural components.Beams are usually long, straight, prismatic members and always subjected forces perpendicular to the axis of the beam 

  • A Shear Force Diagram (SFD) indicates how a force applied perpendicular to the axis (i.e., parallel to cross-section) of a beam is transmitted along the length of that beam.
  • A Bending Moment Diagram (BMD) will show how the applied loads to a beam create a moment variation along the length of the beam.

Types of Supports (a) Roller Support – resists vertical forces only 

 

s2

s

(b) Hinge support or pin connection – resists horizontal and vertical forces

s3

s4

  • Hinge and roller supports are called as simple supports

(c) Fixed support or built-in end

s5

s6

  • The distance between two supports is known as “span”.

Types of beams : Beams are classified based on the type of supports: (1) Simply supported beam: A beam with two simple supports

s7

(2) Cantilever beam: Beam fixed at one end and free at other

s8

(3) Overhanging beam

s9

(4) Continuous beam: More than two supports

s10

 

Shear Force

Shear force has a tendency to slide the surface, it acts parallel to surface.

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Only for distributed load not for point load.

Bending Moment

Any moment produced by forces acting on the beam must be balance by an equal opposite moment produced by internal forces acting in beam at the section. This moment is called bending moment. 

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Only for distributed and concentrated load not for couple.

  • The necessary internal forces to keep the segment of the beam in equilibrium are

s11

Differential equations of equilibrium

s13

Sign Conventions :

s14  

So the differential equations would be:

 

s15

16

From equation S19 ,we can write

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From equation S20,we can write

s18

Statically Determinate Beam

A beam is said to be statically determinate if all its reaction components can be calculated by applying three conditions of static equilibrium i.e.,

image001

 

Statically Indeterminate Beam

When the number of unknown reaction components exceeds the static conditions of equilibrium, the beam is said to be statically indeterminate.

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