Strength of Materials 17ME34
Module 3
Bending Moment and Shear Force
Objectives:
Determine the shear force, bending moment and draw shear force and bending moment diagrams, describe
behaviour of beams under lateral loads. Stresses induced in beams, bending equation derivation & Deflection
behaviour of beams
Learning Structure
• 3.1 Types Of Beams
• 3.2 Shear Force
• 3.3 Bending Moment
• 3.4 Shear Force Diagram And Bending Moment
• 3.5 Relations Between Load, Shear And Moment
• 3.6 Problems
• 3.7 Pure Bending
• 3.8 Effect Of Bending In Beams
• 3.9 Assumptions Made In Simple Bending Theory
• 3.10 Problems
• 3.11 Deflection Of Beams
• Outcomes
• Further Reading
DEPARTMENT OF MECHANICAL ENGINEERING, ATMECE, MYSURU 54
, Strength of Materials 17ME34
3.1 TYPES OF BEAMS
a) Simple Beam
A simple beam is supported by a hinged support at one end and a roller support at the other end.
b) Cantilever beam
A cantilever beam is supported at one end only by a fixed support.
c) Overhanging beam.
An overhanging beam is supported by a hinge and a roller support with either or both ends
extending beyond the supports.
Note: All the beams shown above are the statically determinate beams.
DEPARTMENT OF MECHANICAL ENGINEERING, ATMECE, MYSURU 54
, Strength of Materials 17ME34
W1
Rax-W1a
Ra
Fig 2 :Shear Force Fig 3 : Bending Moment
Consider a simply supported beam subjected to loads W1 and W2. Let RAand RB be the
reactions at supports. To determine the internal forces at C pass a section at C. The effects of
RA and W1 to the left of section are shown in Fig (b) and (c). In each case the effect of applied
load has been transferred to the section by adding a pair of equal and opposite forces at that
section. Thus at the section, moment M = (W1a-Rax) and shear force F = (RA-W1), exists. The
moment M which tend to bends the beam is called bending moment and F which tends to shear
the beam is called shear force.
Thus the resultant effect of the forces at one side of the section reduces to a single force and a
couple which are respectively the vertical shear and the bending moment at that section.
Similarly, if the equilibrium of the right hand side portion is considered, the loading is reduced
to a vertical force and a couple acting in the opposite direction. Applying these forces to a free
body diagram of a beam segment, the segments to the left and right of section are held in
equilibrium by the shear and moment at section.
Thus the shear force at any section can be obtained by considering the algebraic sum of all the
vertical forces acting on any one side of the section
Bending moment at any section can be obtained by considering the algebraic sum of all the
moments of vertical forces acting on any one side of the section.
3.2 Shear Force
It is a single vertical force developed internally at any point on the beam to balance the
external vertical forces and keep the point in equilibrium. It is therefore equal to algebraic sum
of all external forces acting to either left or right of the section.
3.3 Bending Moment
It is a moment developed internally at each point in a beam that balances the external
moments due to forces and keeps the point in equilibrium. It is the algebraic sum of moments
to section of all forces either on left or on right of the section.
DEPARTMENT OF MECHANICAL ENGINEERING, ATMECE, MYSURU 54
Module 3
Bending Moment and Shear Force
Objectives:
Determine the shear force, bending moment and draw shear force and bending moment diagrams, describe
behaviour of beams under lateral loads. Stresses induced in beams, bending equation derivation & Deflection
behaviour of beams
Learning Structure
• 3.1 Types Of Beams
• 3.2 Shear Force
• 3.3 Bending Moment
• 3.4 Shear Force Diagram And Bending Moment
• 3.5 Relations Between Load, Shear And Moment
• 3.6 Problems
• 3.7 Pure Bending
• 3.8 Effect Of Bending In Beams
• 3.9 Assumptions Made In Simple Bending Theory
• 3.10 Problems
• 3.11 Deflection Of Beams
• Outcomes
• Further Reading
DEPARTMENT OF MECHANICAL ENGINEERING, ATMECE, MYSURU 54
, Strength of Materials 17ME34
3.1 TYPES OF BEAMS
a) Simple Beam
A simple beam is supported by a hinged support at one end and a roller support at the other end.
b) Cantilever beam
A cantilever beam is supported at one end only by a fixed support.
c) Overhanging beam.
An overhanging beam is supported by a hinge and a roller support with either or both ends
extending beyond the supports.
Note: All the beams shown above are the statically determinate beams.
DEPARTMENT OF MECHANICAL ENGINEERING, ATMECE, MYSURU 54
, Strength of Materials 17ME34
W1
Rax-W1a
Ra
Fig 2 :Shear Force Fig 3 : Bending Moment
Consider a simply supported beam subjected to loads W1 and W2. Let RAand RB be the
reactions at supports. To determine the internal forces at C pass a section at C. The effects of
RA and W1 to the left of section are shown in Fig (b) and (c). In each case the effect of applied
load has been transferred to the section by adding a pair of equal and opposite forces at that
section. Thus at the section, moment M = (W1a-Rax) and shear force F = (RA-W1), exists. The
moment M which tend to bends the beam is called bending moment and F which tends to shear
the beam is called shear force.
Thus the resultant effect of the forces at one side of the section reduces to a single force and a
couple which are respectively the vertical shear and the bending moment at that section.
Similarly, if the equilibrium of the right hand side portion is considered, the loading is reduced
to a vertical force and a couple acting in the opposite direction. Applying these forces to a free
body diagram of a beam segment, the segments to the left and right of section are held in
equilibrium by the shear and moment at section.
Thus the shear force at any section can be obtained by considering the algebraic sum of all the
vertical forces acting on any one side of the section
Bending moment at any section can be obtained by considering the algebraic sum of all the
moments of vertical forces acting on any one side of the section.
3.2 Shear Force
It is a single vertical force developed internally at any point on the beam to balance the
external vertical forces and keep the point in equilibrium. It is therefore equal to algebraic sum
of all external forces acting to either left or right of the section.
3.3 Bending Moment
It is a moment developed internally at each point in a beam that balances the external
moments due to forces and keeps the point in equilibrium. It is the algebraic sum of moments
to section of all forces either on left or on right of the section.
DEPARTMENT OF MECHANICAL ENGINEERING, ATMECE, MYSURU 54
