Short Notes: Bending ,Shear & Combined Stresses | Short Notes for Civil Engineering - Civil Engineering (CE) PDF Download

 Page 1


Bending ,Shear and Combined Stresses 
Bending stress and shear stress distribution are classified in the following groups 
Bending Moment in Beam:
Transverse loads or lateral loads: Forces or moments having their vectors 
perpendicular to the axis of the bar.
Classification of structural members
• Axially loaded bars: Supports forces having their vectors directed along the 
axis of the bar.
• Bar in tension: Supports torques having their moment vectors directed along 
the axis.
• Beams
° Subjected to lateral loads.
° Beams undergo bending (flexure) because of lateral loads.
When beam is subjected to a bending moment or bent there are induced 
longitudinal or bending stress in cross-section.
Bending Stress in beam:
Page 2


Bending ,Shear and Combined Stresses 
Bending stress and shear stress distribution are classified in the following groups 
Bending Moment in Beam:
Transverse loads or lateral loads: Forces or moments having their vectors 
perpendicular to the axis of the bar.
Classification of structural members
• Axially loaded bars: Supports forces having their vectors directed along the 
axis of the bar.
• Bar in tension: Supports torques having their moment vectors directed along 
the axis.
• Beams
° Subjected to lateral loads.
° Beams undergo bending (flexure) because of lateral loads.
When beam is subjected to a bending moment or bent there are induced 
longitudinal or bending stress in cross-section.
Bending Stress in beam:
M M
• I is Moment of Inertia about Neutral Axis
• Note that a positive bending moment M causes negative (compressive) stress 
above the neutral axis and positive ( tensile) stress below the neutral axis
Equation of Pure Bending:
v ” 1 ~ R
Assumptions:
1. The material of the beam is homogeneous and isotropic.
2. The value of Young's Modulus of Elasticity is same in tension and 
compression.
3. The transverse sections which were plane before bending, remain plane after 
bending also.
4. The beam is initially straight and all longitudinal filaments bend into circular 
arcs with a common centre of curvature.
5. The radius of curvature is large as compared to the dimensions of the cross- 
section.
6. Each layer of the beam is free to expand or contract, independently of the 
layer, above or below it.
Modulus of Section
• Elastic section modulus is defined as Z(S) = I / y, where I is the second 
moment of area (or Izz moment of inertia) and y is the distance from the 
neutral axis to any given fibre.
• Section modulus is a geometric property for a given cross-section used in the 
design of beams or flexural members.
Page 3


Bending ,Shear and Combined Stresses 
Bending stress and shear stress distribution are classified in the following groups 
Bending Moment in Beam:
Transverse loads or lateral loads: Forces or moments having their vectors 
perpendicular to the axis of the bar.
Classification of structural members
• Axially loaded bars: Supports forces having their vectors directed along the 
axis of the bar.
• Bar in tension: Supports torques having their moment vectors directed along 
the axis.
• Beams
° Subjected to lateral loads.
° Beams undergo bending (flexure) because of lateral loads.
When beam is subjected to a bending moment or bent there are induced 
longitudinal or bending stress in cross-section.
Bending Stress in beam:
M M
• I is Moment of Inertia about Neutral Axis
• Note that a positive bending moment M causes negative (compressive) stress 
above the neutral axis and positive ( tensile) stress below the neutral axis
Equation of Pure Bending:
v ” 1 ~ R
Assumptions:
1. The material of the beam is homogeneous and isotropic.
2. The value of Young's Modulus of Elasticity is same in tension and 
compression.
3. The transverse sections which were plane before bending, remain plane after 
bending also.
4. The beam is initially straight and all longitudinal filaments bend into circular 
arcs with a common centre of curvature.
5. The radius of curvature is large as compared to the dimensions of the cross- 
section.
6. Each layer of the beam is free to expand or contract, independently of the 
layer, above or below it.
Modulus of Section
• Elastic section modulus is defined as Z(S) = I / y, where I is the second 
moment of area (or Izz moment of inertia) and y is the distance from the 
neutral axis to any given fibre.
• Section modulus is a geometric property for a given cross-section used in the 
design of beams or flexural members.
z - 1 M - a
1 y
M = < j => M = a x z
o Modulus of the section :
z
hd2
6
• Circular section:
7 =
bds
12
7 =
° Modulus of section :
z = — d3 
32
Page 4


Bending ,Shear and Combined Stresses 
Bending stress and shear stress distribution are classified in the following groups 
Bending Moment in Beam:
Transverse loads or lateral loads: Forces or moments having their vectors 
perpendicular to the axis of the bar.
Classification of structural members
• Axially loaded bars: Supports forces having their vectors directed along the 
axis of the bar.
• Bar in tension: Supports torques having their moment vectors directed along 
the axis.
• Beams
° Subjected to lateral loads.
° Beams undergo bending (flexure) because of lateral loads.
When beam is subjected to a bending moment or bent there are induced 
longitudinal or bending stress in cross-section.
Bending Stress in beam:
M M
• I is Moment of Inertia about Neutral Axis
• Note that a positive bending moment M causes negative (compressive) stress 
above the neutral axis and positive ( tensile) stress below the neutral axis
Equation of Pure Bending:
v ” 1 ~ R
Assumptions:
1. The material of the beam is homogeneous and isotropic.
2. The value of Young's Modulus of Elasticity is same in tension and 
compression.
3. The transverse sections which were plane before bending, remain plane after 
bending also.
4. The beam is initially straight and all longitudinal filaments bend into circular 
arcs with a common centre of curvature.
5. The radius of curvature is large as compared to the dimensions of the cross- 
section.
6. Each layer of the beam is free to expand or contract, independently of the 
layer, above or below it.
Modulus of Section
• Elastic section modulus is defined as Z(S) = I / y, where I is the second 
moment of area (or Izz moment of inertia) and y is the distance from the 
neutral axis to any given fibre.
• Section modulus is a geometric property for a given cross-section used in the 
design of beams or flexural members.
z - 1 M - a
1 y
M = < j => M = a x z
o Modulus of the section :
z
hd2
6
• Circular section:
7 =
bds
12
7 =
° Modulus of section :
z = — d3 
32
Beams of uniform strength
• The beam is said to be in uniform strength if the maximum bending stress is 
constant across the varying section along its length.
• Generally, beams are having the uniform cross-section throughout their 
length. When a beam is loaded, there is a variation in bending moment from 
section to section along the length. The stress in extreme outer fibre (top and 
bottom) also varies from section to section along their length. The extreme 
fibres can be loaded to the maximum capacity of permissible stress
(say pm ax). but they are loaded to less capacity. Hence, in beams of uniform 
cross section, there is a considerable waste of materials
• When a beam is suitably designed such that the extreme fibres are loaded to 
the maximum permissible stress pm ax by varying the cross section it will be 
known as a beam of uniform strength.
Shearing Stress
• Shearing stress on a layer JK of beam at distance y from neutral axis.
uC i
4
y
.... r
H — dx—
Shearing stress on a beam
Where,
VAy
~lb~
• V = Shearing force
Ay=
First moment of area
VO
7 b
Shear stress in Rectangular Beam
• Suppose, we have to determine the shear stress at the longitudinal layer 
having y distance from neutral axis.
0 = b
>i +
Page 5


Bending ,Shear and Combined Stresses 
Bending stress and shear stress distribution are classified in the following groups 
Bending Moment in Beam:
Transverse loads or lateral loads: Forces or moments having their vectors 
perpendicular to the axis of the bar.
Classification of structural members
• Axially loaded bars: Supports forces having their vectors directed along the 
axis of the bar.
• Bar in tension: Supports torques having their moment vectors directed along 
the axis.
• Beams
° Subjected to lateral loads.
° Beams undergo bending (flexure) because of lateral loads.
When beam is subjected to a bending moment or bent there are induced 
longitudinal or bending stress in cross-section.
Bending Stress in beam:
M M
• I is Moment of Inertia about Neutral Axis
• Note that a positive bending moment M causes negative (compressive) stress 
above the neutral axis and positive ( tensile) stress below the neutral axis
Equation of Pure Bending:
v ” 1 ~ R
Assumptions:
1. The material of the beam is homogeneous and isotropic.
2. The value of Young's Modulus of Elasticity is same in tension and 
compression.
3. The transverse sections which were plane before bending, remain plane after 
bending also.
4. The beam is initially straight and all longitudinal filaments bend into circular 
arcs with a common centre of curvature.
5. The radius of curvature is large as compared to the dimensions of the cross- 
section.
6. Each layer of the beam is free to expand or contract, independently of the 
layer, above or below it.
Modulus of Section
• Elastic section modulus is defined as Z(S) = I / y, where I is the second 
moment of area (or Izz moment of inertia) and y is the distance from the 
neutral axis to any given fibre.
• Section modulus is a geometric property for a given cross-section used in the 
design of beams or flexural members.
z - 1 M - a
1 y
M = < j => M = a x z
o Modulus of the section :
z
hd2
6
• Circular section:
7 =
bds
12
7 =
° Modulus of section :
z = — d3 
32
Beams of uniform strength
• The beam is said to be in uniform strength if the maximum bending stress is 
constant across the varying section along its length.
• Generally, beams are having the uniform cross-section throughout their 
length. When a beam is loaded, there is a variation in bending moment from 
section to section along the length. The stress in extreme outer fibre (top and 
bottom) also varies from section to section along their length. The extreme 
fibres can be loaded to the maximum capacity of permissible stress
(say pm ax). but they are loaded to less capacity. Hence, in beams of uniform 
cross section, there is a considerable waste of materials
• When a beam is suitably designed such that the extreme fibres are loaded to 
the maximum permissible stress pm ax by varying the cross section it will be 
known as a beam of uniform strength.
Shearing Stress
• Shearing stress on a layer JK of beam at distance y from neutral axis.
uC i
4
y
.... r
H — dx—
Shearing stress on a beam
Where,
VAy
~lb~
• V = Shearing force
Ay=
First moment of area
VO
7 b
Shear stress in Rectangular Beam
• Suppose, we have to determine the shear stress at the longitudinal layer 
having y distance from neutral axis.
0 = b
>i +
T m a x —
:s V
2 A
T m ;ix —
1 .5 t
avR
Rectangular beam
Circular Beam
• Centre of gravity of semi-circle lies at distance from centre or base line. As it 
is symmetrical above neutral axis, hence at neutral axis shear stress will be 
maximum.
T - V
r - y j
3/
0 = A v
7rr* 4)
3t t
b = 2r
- r
3
‘ tnx
4 V 
3 A
• F o r r _ s u b s ititu tin s v = 0
r j x o j 4
l =
-d*
64
4 V 4
3
Shears Stress in Hollow Circular Cross-Section