Short Notes Shear Force and Bending Moment Diagrams - Strength of Materials

 Page 1


Shear Force and Bending Moment Diagrams  
A 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
i Rb
(b) Hinge support or pin connection - resists horizontal and vertical forces
Page 2


Shear Force and Bending Moment Diagrams  
A 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
i Rb
(b) Hinge support or pin connection - resists horizontal and vertical forces
7777777
A ____
t R'y
Resists horizontal 
and vertical forces
• Hinge and roller supports are called as simple supports
(c) Fixed support or built-in end
and vertical forces 
and moment
• 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
S im p ly supported beams
(2) Cantilever beam: Beam fixed at one end and free at other
?TTTTT
T r
_____________
(3) Overhanging beam
Page 3


Shear Force and Bending Moment Diagrams  
A 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
i Rb
(b) Hinge support or pin connection - resists horizontal and vertical forces
7777777
A ____
t R'y
Resists horizontal 
and vertical forces
• Hinge and roller supports are called as simple supports
(c) Fixed support or built-in end
and vertical forces 
and moment
• 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
S im p ly supported beams
(2) Cantilever beam: Beam fixed at one end and free at other
?TTTTT
T r
_____________
(3) Overhanging beam
L
(4) Continuous beam: More than two supports
Shear Force
The resultant force which is in upward 
direction and is towards the L.H.S of the 
X-sectlon is +ve Shear Force
The resultant force which is in the downward 
direction and is towards the R.H.S of the 
X-section is +ve Shear Force.
The resultant force which are in the downward 
direction and is on the L.H.S of the X-section 
is -ve Shear Force.
The resultant force which are in upward 
direction and is on the R.H.S of the 
X-sectlon Is -ve Shear Force.
Shear force has a tendency to slide the surface, it acts parallel to surface.
Y.F = 0
— * xm
Page 4


Shear Force and Bending Moment Diagrams  
A 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
i Rb
(b) Hinge support or pin connection - resists horizontal and vertical forces
7777777
A ____
t R'y
Resists horizontal 
and vertical forces
• Hinge and roller supports are called as simple supports
(c) Fixed support or built-in end
and vertical forces 
and moment
• 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
S im p ly supported beams
(2) Cantilever beam: Beam fixed at one end and free at other
?TTTTT
T r
_____________
(3) Overhanging beam
L
(4) Continuous beam: More than two supports
Shear Force
The resultant force which is in upward 
direction and is towards the L.H.S of the 
X-sectlon is +ve Shear Force
The resultant force which is in the downward 
direction and is towards the R.H.S of the 
X-section is +ve Shear Force.
The resultant force which are in the downward 
direction and is on the L.H.S of the X-section 
is -ve Shear Force.
The resultant force which are in upward 
direction and is on the R.H.S of the 
X-sectlon Is -ve Shear Force.
Shear force has a tendency to slide the surface, it acts parallel to surface.
Y.F = 0
— * xm
V—qdx— (V +dV)=0
d r
dx
~9
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.
v \ \ / = 0
— M — qdx
dx
z
~ {V + d V )d x+ M + dm = 0
dU
Fix
= V => Ma — M A = f V dx
Only for distributed and concentrated load not for couple.
• The necessary internal forces to keep the segment of the beam in equilibrium 
are
£ F X = 0 =>P 
Z F y = 0 =>V 
Z F z = 0 = > M
Resultant moment on the L.H.S of 
the X-section is C.W, then it is a 
positive B.M
A
Resultant moment on the R.H.S postion 
of the X-section is C C W. then it may be 
considered as positive B.M
Page 5


Shear Force and Bending Moment Diagrams  
A 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
i Rb
(b) Hinge support or pin connection - resists horizontal and vertical forces
7777777
A ____
t R'y
Resists horizontal 
and vertical forces
• Hinge and roller supports are called as simple supports
(c) Fixed support or built-in end
and vertical forces 
and moment
• 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
S im p ly supported beams
(2) Cantilever beam: Beam fixed at one end and free at other
?TTTTT
T r
_____________
(3) Overhanging beam
L
(4) Continuous beam: More than two supports
Shear Force
The resultant force which is in upward 
direction and is towards the L.H.S of the 
X-sectlon is +ve Shear Force
The resultant force which is in the downward 
direction and is towards the R.H.S of the 
X-section is +ve Shear Force.
The resultant force which are in the downward 
direction and is on the L.H.S of the X-section 
is -ve Shear Force.
The resultant force which are in upward 
direction and is on the R.H.S of the 
X-sectlon Is -ve Shear Force.
Shear force has a tendency to slide the surface, it acts parallel to surface.
Y.F = 0
— * xm
V—qdx— (V +dV)=0
d r
dx
~9
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.
v \ \ / = 0
— M — qdx
dx
z
~ {V + d V )d x+ M + dm = 0
dU
Fix
= V => Ma — M A = f V dx
Only for distributed and concentrated load not for couple.
• The necessary internal forces to keep the segment of the beam in equilibrium 
are
£ F X = 0 =>P 
Z F y = 0 =>V 
Z F z = 0 = > M
Resultant moment on the L.H.S of 
the X-section is C.W, then it is a 
positive B.M
A
Resultant moment on the R.H.S postion 
of the X-section is C C W. then it may be 
considered as positive B.M
A
negative B.M ¦ negative B.M
A
Differential equations of equilibrium
P(x)kg per unit length
1111 1 i ^
rrmr
\X—
So the differential equations would be:
1 4 AV d V
4*->0 A X “ X
7. AM dM
l i m — = - i - = ~ v
A t— >0 A x “ x
,we can write 
V d - V c = - \ Pdx 
From equation
d y
d x
-M
,we can write
M D - M c = - \ V d x