1. Mechanics of solidsMechanics of solids
Torsion,
Bending moment and shear forceBending moment and shear force
Dr. Rajesh K. N.
Assistant Professor in Civil EngineeringAssistant Professor in Civil Engineering
Govt. College of Engineering, Kannur
1
2. Module IIModule II
Torsion - torsion of circular elastic bars - statically
indeterminate problems - torsion of inelastic circular bars
Axial force, shear force and bending moment -
diagrammatic conventions for supports and loading, axialdiagrammatic conventions for supports and loading, axial
force, shear force and bending moment diagrams - shear
force and bending moments by integration and by
singularity functions
Dept. of CE, GCE Kannur Dr.RajeshKN
2
4. Assumptions in torsion theory for circular shaftsAssumptions in torsion theory for circular shafts
⢠Material is uniform throughout
⢠Shaft remains circular after loading
⢠Plane sections remain plain after loading
T i i if h h⢠Twist is uniform throughout
⢠Distance between any two normal sections remain the same after loading
⢠Stresses are within elastic limit⢠Stresses are within elastic limit
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4
5. T
Ď Î¸
mâ
r
T
Ď Î¸
m
O
r
T
L
Any cross-section
Any radial distance r
mm L rĎ Î¸â˛ = =
Angle of twist per unit length
L
θ
â
mm L rĎ Î¸
G
Ď
Ď =Shear strain
L
θ
O
r
rĎ Î¸
â´ =
G
rθ
Ď =But
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5
G L
â´
L
Ď =But
6. T
Torsion equation
R
T
θ
r
Ď
θ
L
T
O
maxG
r L R
Ď Î¸ Ď
â´ = =
r
G L
Ď Î¸
=
Aδ
maxr
R
Ď
Ďâ =
. .T A rδ Ď Î´=
2max max
.
r
T A r r A
R R
Ď Ď
δ δ δ= =
Torsional moment on the elemental area =
2max
T T r A
R
Ď
δ δ= =⍠âŤ
R R
Total torsional moment on the section,
A A
R⍠âŤ
2max
r A
R
Ď
δ= ⍠max
J
R
Ď
=
,
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A
R R
Polar moment of inertiaJ â
7. T Ď T GθmaxT
J R
Ď
= T G
J r L
Ď Î¸
= =
T
GJ L
θ
= GJ Torsional rigidity
GJ L
maxT TĎ J R Torsional section modulusmax
max
J R J R
Ď= â = J R Torsional section modulus
4
dĎ
ZZ XX YYJ I I I= = +Polar moment of inertia
32
dĎ
=
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7
8. Power transmittedo e t a s tted
W T θ=
Work done (per second) by torque T making a twist θ1 /second
1.W T θ=
1.P T θ=i.e., Power transmitted
2
60
nT
P
Ď
=If n is the rotation per minute,
k
1 Watt 1 Joule/Second = 1 Nm/s= =
1 HP 0.75 kW=
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8
9. Problem 1: A hollow shaft is to transmit a power of 300 kW at 80 rpm.
If the shear stress is not to exceed 60 MN/m2 and internal diameter is
0.6 of external diameter, find the internal and external diameters
assuming that the maximum torque is 1.4 times the mean torque.g q q
300 kWP =
2
60
nT
P
Ď
=80 rpmn =
60
2
mean
P
T T
nĎ
â´ = =
3
60 300 10
35809.862 Nm 35.81 kNm
2 80
meanT
Ď
Ă Ă
= = =
Ă
max 1.4 meanT Tâ´ = 1.4 35.81 50.134 kNm= Ă =
maxT
J R
Ď
= ( ) ( )( )44 4 4
0.6
32 32
J D d D D
Ď Ď
= â = â ( )
4
0.8704
32
DĎ
=
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J R 32 32
11. Problem 2: A solid circular shaft is to transmit 75 kW power at 200 rpm.
If the shear stress is not to exceed 50 MPa, and the twist is not to exceed
10 in 2 m length of shaft, find the diameter of shaft. G = 100 GPa.
75 kWP =
2
60
nT
P
Ď
=200 rpmn =
3
60 60 75 10
3581 Nm
2 2 200
P
T
Ă Ă
= = =
Ă2 2 200nĎ Ď Ă
maxT GĎ Î¸ 4
D
J
Ď 2
50 N mmĎ =max
J R L
= =
32
J = max 50 N mmĎ =
6
4
3581 50 10
2
32
DDĎ
Ă
=
â â
â â
â â
0.0714 mDâ =maxT
J R
Ď
= â
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11
32â â
12. T G
J L
θ
=
9
4
3581 100 10 1 180
2
32
D
Ď
Ď
Ă Ă Ă
â =
â â
â â
â â
0.0804 mDâ =
32â â
Required diameter of shaft is the greater of the two values
0.0804 m 80.4 mmDâ´ = =
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12
13. Problem 3: A steel bar of 20 mm diameter and 450 mm length fails at a
torque of 800 Nm. What is the modulus of rupture of this steel in
torsion?
Modulus of rupture in torsion is the maximum shear
stress (on the surface) at failure.
T
R J
Ď
=
Hence, modulus of rupture
TR
J
Ď = where T is the torque at failure.
3
2800 10 10
509N
Ă Ă
J
2
4
800 10 10
509N mm
20
32
Ď
Ď
= =
â âĂ
â â
â â
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14. Problem 4: A solid brass rod AB (G=39 Gpa, 30 mm dia, 250 mm length) is
b d d lid l i i d BC (G 27 G 36 di 320 l h)bonded to solid aluminium rod BC (G=27 Gpa, 36 mm dia, 320 mm length).
Determine the angles of twist at A and B.
T G TL
J L GJ
θ
θ= â =
A
BC
3
3
180 10 320
27 10 164895.92
Bθ
Ă Ă
=
Ă Ă
0
0.0129 rad 0.74= =
Torque, T
180 Nm
4
436
164895.92mm
32
BCJ
Ď Ă
= =
3
3
180 10 250
39 10 79521 56
ABθ
Ă Ă
=
Ă Ă
0
0.0145 rad 0.831= =
4
430
79521.56mm
32
ABJ
Ď Ă
= =
39 10 79521.56Ă Ă
0 00
0.74 0.83 . 711 1 5ACθ = + =
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15. Problem 5: Two solid brass rods (G=39 Gpa) AB (30mm dia, 1.2m length) & BC
(40 di 1 8 l h) li d i h h D i h(40mm dia, 1.8m length), are applied with torques as shown. Determine the
angles of twist between i) A and B; ii) A and C.
T G TL
J L GJ
θ
θ= â =
3
3
400 10 1200
39 10 79521.56
BAθ
Ă Ă
=
Ă Ă
0.1548 rad=
3
3
800 10 1800
39 10 251327.41
CBθ
â Ă Ă
=
Ă Ă
0.1469 rad= â
4
430
79521.56mm
32
ABJ
Ď Ă
= =θ θ θ= +
4
440
251327.41 mmBCJ
Ď Ă
= =
32
AB
CA BA CBθ θ θ= +
0.1548 0.1469 0.0079 radCAθ = â =
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251327.41 mm
32
BCJ
16. Statically indeterminate shaftsy
A
B
T T
B
C
Torque, T
A BT T T= + (1)
A
B
0Aθ = with respect to B
0θ θâ =
Torque, T
B
C
0CB ACθ θ =
0B BC A CAT L T L
G J G J
â â =Torque, T
Reactive
torque, TA
Reactive
torque, TB
BC BC CA CAG J G J
(2)
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Solve (1) and (2) for &A BT T
17. Torsion of inelastic circular bars
⢠Linear elastic range: Stress strain curve is a straight line
(Torsion of circular bars beyond elastic range)
g g
Stress
Strain
⢠Corresponding shear stress distribution in the shaft:
Ď
T
r
J
Ď =
maxĎ
R Entire cross-section is in the
elastic range
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18. Materials in the inelastic ranges
⢠Inelastic: Nonlinear stress strain curve
g
Stress
Elastic rebound
on unloading
⢠Elastic plastic : Initially in the elastic range then fully plastic
Strain
⢠Elastic-plastic : Initially in the elastic range, then fully plastic
Stress
Elastic rebound
on unloading
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S
Strain
19. Shear stress-strain curve Corresponding shear stress
distribution in the shaft
ss
maxĎStres
Strain
R
ss
maxĎ
Stres
Strain
R
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20. maxĎ maxĎFor any stress distribution,
. .dT dA rĎ=
T dT r dAĎ= =⍠âŤ
R R
y ,
maxĎ
R
.
A A
T dT r dAĎ⍠âŤ
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21. Problem 1:
A solid steel shaft of 24 mm diameter is so severely twisted that only an 8 mm
diameter inner core remains elastic, while the rest of the diameter goes to
inelastic range. If the material is elastic-plastic with shear stress-strain diagram
h h h id l d id l i h ill i has shown, what are the residual stress and residual twist that will remain at the
surface? Take G = 80 GPa.
Ď
160 MPa
4 mm
12 mm
0.002 Îł Ď=
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22. The inner core is in the elastic rangeThe inner core is in the elastic range. 160 MPa
12 mm4 mm
It is required to find the applied torque.
âŤ
R R
⍠âŤ
Applied torque,
. .
A
T r dAĎⲠ= âŤ
2
0 0
. .2 . .2 .r r dr r drĎ Ď Ď Ď= =⍠âŤ
4 12
160â â
( )
4 12
2 2
0 4
160
.2 . 160 .2 .
4
r r dr r drĎ Ďâ â
= +â â
â â
⍠âŤ
( ) 3 3
16 558 10 Nmm 574 10 Nmm= + Ă = Ă
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23. Residual stresses on rebound (removal of torque)
⢠Consider the external torque is removed after a portion of the cross-section
has gone to inelastic range
( q )
has gone to inelastic range.
⢠After the torque has been removed completely, the inner portion has a
tendency to rotate back to original position but the outer portion which has atendency to rotate back to original position, but the outer portion, which has a
permanent rotation, prevents this.
⢠Outer portion has a tendency to stay in the permanently set position, butOuter portion has a tendency to stay in the permanently set position, but
the inner portion, which has a tendency to rotate back to original position,
prevents this.
⢠Hence, stresses are remaining in the shaft even after the removal of torsion.
⢠The rebound is elastic.
⢠If the deformation of the outer portion was elastic, shear stress would have
reached a value of Ďâ at the surface.
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24. ⢠The stress recovery at the surface is Ďmax .
⢠Hence, as a result of inner portion applying a torsion on the outer portion, a
stress remains at the surface, equal to Ďâ - Ďmax , opposite in direction to the
li d t i it t th di ti fapplied torque. i.e., opposite to the direction of Ďmax .
⢠Similarly, all the inner regions at various radial distances have residual
stressesstresses.
⢠These residual stresses are obtained as the difference between elastic-plastic
stress distribution and the elastic stress distribution of rebound as shown instress distribution and the elastic stress distribution of rebound, as shown in
figure.
Elastic rebound
Ďâ˛E g : Elastic plastic material
Residual stress
maxĎ
E.g.: Elastic-plastic material
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25. To find residual stresses on an elastic rebound, (i.e., on removal of the
T RⲠ3
2574 10 12Ă Ă
torque)
Shear stress on the surface T R
J
ĎⲠ= 2
4
574 10 12
211 N mm
24
32
Ď
Ă Ă
= =
â âĂ
â â
â â
(considering elastic rebound)
Hence, residual shear stress on the surface
2
211 160 51 N mm= â =
maxĎ Ďâ˛= â
160 MPa
Residual
stress
211 MPa
12 mm4 mm
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26. 0.006Îł ĎⲠâ˛= =
Residual twist on rebound
Strain variation is linear along the radius. 4 0.002Ď =
Residual twist on rebound
θ
12mmR =
4mmmax max
L RG
θ Ď
=At the initiation of yield,
maxT T=
When T increases further to Tâ,
L
θ
increases beyond max
, ot
L L
θ θⲠStrain variation
L L L
L rG
θ Ď
=After yielding , is invalid from r=4 to 12.Note:
Final twist
(after yielding,
when T=Tâ)
θâ˛
i.e., .
L RG
θ ĎⲠâ˛
â But, r
L r R
θ Ď ĎⲠâ˛
= =
On elastic rebound, recovered twist Elastic Rebound
(recovery of twist)
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T
L GJ
θ â˛
=
Twist of a radial line
(recovery of twist)
27. To find residual twist on an elastic rebound,
3
0.002
4 10
0.5 rad
m
mâ
= =
Ă
r
L r
θ Ďâ˛
=
Final twist per unit length (after yielding,
when T=Tâ)
L r
3
rad m
160
4
m 0.5 ra
10
d
8
m
0
= =
Ă Ă
or,
L rG
θ Ďâ˛
=
4 1080L rG
L rG
θ Ď
=âľ is valid from r=0 to 4.
Elastic twist recovered (considering elastic rebound)
4
9 2 12 4
574Nm
24
80 10 N m 10 m
Ď â
=
â âĂ
Ă Ă Ăâ â
( g )
T
L GJ
θ â˛
= 0.22 rad m=
Hence, residual twist 0.5 0.22 0.28rad m= â =
80 10 N m 10 m
32
Ă Ă Ăâ â
â â
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28. Problem 2:
A solid circular shaft 1 2 m long and 50 mm diameter is subjected to a torque ofA solid circular shaft, 1.2 m long and 50 mm diameter is subjected to a torque of
4.6 kNm. Assuming the shaft is made of an elastoplastic material with yield
strength in shear of 150MPa and G = 80 Gpa, what are the radius of the elastic
core and angle of twist of the shaft? Also, what are the residual stresses andcore and angle of twist of the shaft? Also, what are the residual stresses and
residual (permanent) angle of twist of the shaft, after removal of torque?
A li d t
150 MPa
25 mmĎ
To find elastic core.
. .
A
T r dAĎⲠ= âŤ
2
0 0
. .2 . .2 .
R R
r r dr r drĎ Ď Ď Ď= =⍠âŤ
Applied torque, 25 mmĎ
( )
25
6 2 2
0
150
4.6 10 N.mm .2 . 150 .2 .r r dr r dr
Ď
Ď
Ď Ď
Ď
â â
Ă = +â â
â â
⍠âŤ
( )3 33
6
2 252
4.6 10 N.mm 150 150
4 3
Ď ĎĎĎ â
Ă = +
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28
15.743 mmĎ =â´
29. To find angle of twist
Îł â˛
To find angle of twist.
(Final angle of twist in the inelastic (plastic) range) ?rĎ =
25 mmR =15.743 mmĎ =
G
L
Ď Î¸
Ď
= (Valid from r = 0 to 15.743 only)
Strain variation along radius
3
150 1200
80 10 15.743
L
G
Ď
θ
Ď
Ă
â´ = =
Ă ĂĎ
3
142.92 10 radâ
= Ă
Final twist
(after yielding,
when T=Tâ)
θâ˛
0
3 0180
142.9 8. 82 0 11
Ď
â
= Ă Ă =
Elastic Rebound
(reco er of t ist)
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29Twist of a radial line
(recovery of twist)
30. To find residual stresses on an elastic rebound, (i.e., on removal of the
T Râ˛
6
4.6 10 25Ă Ă
torque)
Shear stress on the surface 2
187 52NT R
J
ĎⲠ= 4
50
32
Ď
=
â âĂ
â â
â â
(considering elastic rebound)
2
187.52N mm=
Hence, residual shear stress on the surface
2
187.52 150 37.52 N mm= â =
maxĎ Ďâ˛= â
150 MPa
Residual
stress
187.52 MPa
25 mm15.743 mm
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31. To find residual twist on an elastic rebound,
Elastic twist recovered (considering elastic rebound)
3
4
9 2 12 4
4.6 10 Nm 1.2m
50
80 10 N m 10 m
Ď â
Ă Ă
=
â âĂ
Ă Ă Ăâ â
T L
GJ
θ
â˛
=
0
0.1124 rad 6.44= =
Hence residual twist 0
8 18 6 4 74 1 4= â =
80 10 N m 10 m
32
Ă Ă Ăâ â
â â
Hence, residual twist 8.18 6.4 74 1. 4= =
Final twist
(after yielding,
h T=Tâ)
θâ˛
when T=T )
Elastic Rebound
(recovery of twist)
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Twist of a radial line
(recovery of twist)
32. Problem 3:
A solid circular steel shaft 0 6 m long and 32 mm diameter has been twistedA solid circular steel shaft 0.6 m long and 32 mm diameter has been twisted
through 60. Steel is elastoplastic with yield strength in shear of 145MPa and G
= 77 Gpa. What is the maximum residual stress in the shaft, after removal of
torque?torque?
To find elastic core.
6 0.1047 rad
180
Ď
θ = à = 145 MPa
G
L
Ď Î¸
Ď
= (Valid from r = 0 to Ď only)
16 mmĎ
10.792 mm=
Ď
3
145 600
10.792 mm
77 10 0.1047
L
G
Ď
Ď
θ
Ă
â´ = = =
Ă Ă
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32
33. Applied torque
To find torque applied.
. .
A
T r dAĎⲠ= âŤ
2
0 0
. .2 . .2 .
R R
r r dr r drĎ Ď Ď Ď= =⍠âŤ
Applied torque,
A 0 0
( )
10.792 16
2 2
0 10 792
145
.2 . 145 .2 .
10.792
r r dr r drĎ Ďâ â
= +â â
â â
⍠âŤ0 10.792â â
( )3 33 2 16 10.7922 10.792
145 145
4 3
ĎĎ â
= +
1148475.884 NmmTâ˛â´ =
4 3
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34. To find residual stresses on an elastic rebound, (i.e., on removal of the
T Râ˛
â˛
1148475.884 Nmm 16Ă
torque)
Shear stress on the surface
2T R
J
ĎⲠ= 4
1148475.884 Nmm 16
32
32
Ď
=
â âĂ
â â
â â
(considering elastic rebound)
2
178.5N mm=
Hence, residual shear stress on the surface
2
178.5 145 33.5 N mm= â =
maxĎ Ďâ˛= â
max
TR
J
Ď= âResidual shear stress at 10.792 mmĎ =
145 MPa
Residual
stress
178.5 MPa
max4
1148475.884 Nmm 10.792
32
Ď
Ď
Ă
= â
â âĂ
â â
16 mm10.792 mm
2
120.4 145 24.6 N mm= â = â
32â â
â â
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Hence the maximum residual shear stress is on the surface.
35. Problem 4:
A hollow circular steel shaft 1 25 m long 60 mm outer diameter and 36 mmA hollow circular steel shaft 1.25 m long, 60 mm outer diameter and 36 mm
inner diameter has been applied with a torque such that the inner surface first
reaches plastic zone, and then torque is removed. Steel is elastoplastic with
yield strength in shear of 145MPa and G = 77 Gpa What is the maximumyield strength in shear of 145MPa and G 77 Gpa. What is the maximum
residual stress in the shaft and residual (permanent) angle of twist, after
removal of torque?
30
Applied torque,
To find torque applied.
. .
A
T r dAĎⲠ= ⍠2
0 0
. .2 . .2 .
R R
r r dr r drĎ Ď Ď Ď= =⍠⍠( )
30
2
18
145 .2 .r drĎ= âŤ
( )( )3 3
2 30 18
145
3
Ď â
=
145 MPa
6428452.551 NmmTâ˛â´ =
( )4 4
60 36Ď Ă â
18 mm 30 mm
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35
( ) 4
60 36
1107449.11 mm
32
J
Ď Ă
= =
36. To find residual stresses on an elastic rebound, (i.e., on removal of the
T R
J
Ď
â˛
Ⲡ=
6428452.551 Nmm 30
1107449 11
Ă
=
torque)
Shear stress on the surface
(considering elastic rebound) J 1107449.11(considering elastic rebound)
2
174.142 N mm=
Hence, residual shear stress on the outer surface
2
174.142 145 29.142 N mm= â =
maxĎ Ďâ˛= â
Ďâ˛
max
TR
J
Ď= âResidual shear stress at 18 mmĎ =
174 142 MPa=
145 MPa
Residual
stress
Ď
max
6428452.551 Nmm 18
1107449.11
Ď
Ă
= â
174.142 MPa
18 mm 30 mm 2
104.49 145 40.51 N mm= â = â
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Hence the maximum residual shear stress is on the inner surface.
37. To find residual twist on an elastic rebound,
Final twist per unit length (after
yielding, when T=Tâ)
L
rG
Ď
θⲠ=
L rG
θ Ď
=âľ is valid from r=0 to 18.
3
145 1250
rad 0.131 rad
18 77 10
Ă
= =
Ă Ă
Elastic twist recovered (considering elastic rebound)
8 77 0
3 2 4
6428452.551 Nmm 1250mm
77 10 N mm 1107449 11mm
Ă
=
Ă Ă
Elastic twist recovered (considering elastic rebound)
T L
GJ
θ
â˛
= 0.094 rad=
Hence, residual twist 0.131 0.09 0.034 7rad= â =
77 10 N mm 1107449.11mmĂ ĂGJ
0
0.037rad 2.1= =
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38. Bending moment and shear force
T f l d
g
Types of loads
Point load
Distributed loadDistributed load
Uniformly distributed load
Uniformly varying loady y g
Couple
T f tTypes of supports
Fixed (built-in or encastre)
Hinged (pinned)Hinged (pinned)
Roller
Guided fixed
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38
Elastic (Spring)
39. Beams
Statically determinate and indeterminate beams
Beams
y
⢠Simply supportedSimply supported
⢠Cantilever
⢠Propped cantilever
⢠Fixed
⢠Continuous
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39
40. Sign conventions
Bending Momentg
S i H iSagging Hogging
Shear Force
Clockwise Anticlockwise
Dept. of CE, GCE Kannur Dr.RajeshKN
40
Clockwise Anticlockwise
41. Shear force and bending moment diagramsShear force and bending moment diagrams
1 Cantilever1. Cantilever
⢠with a single load at the free end
⢠with several point loadswith several point loads
⢠with UD load over full span
⢠with UD load over part spanp p
⢠with a couple
⢠with combination of loads
Dept. of CE, GCE Kannur Dr.RajeshKN
41
44. Si l t d b
⢠with a single point load at the centre
Simply supported beams
⢠with a single point load at the centre
⢠with a single eccentric point load
⢠with several point loadsp
⢠with UD load over full span
⢠with UD load over part span
h b f l d⢠with combination of loads
⢠with a couple
Dept. of CE, GCE Kannur Dr.RajeshKN
44
45. C
P
( ) 2
SFD
C
A B
2
Pâ
(+)
(-)SFD 2
4
PL
A B
Dept. of CE, GCE Kannur Dr.RajeshKN
45
BMD C
48. 60 kN 50 kN
1
A D C B
1m
3m
6m
x
Dept. of CE, GCE Kannur Dr.RajeshKN
48
49. Simply supported beam with overhangp y pp g
⢠Overhang on one side with point loadOverhang on one side with point load
⢠Overhang on one side with UD load over full span
⢠Overhang on both sides with point loadsg
⢠Overhang on both sides with UD load over full span
Dept. of CE, GCE Kannur Dr.RajeshKN
49
50. P
ba
x
60 kN
30 kN/m
5 kN/m25 kNm
A DCB E F
1m 2m1m 1.5m 2.5m
Dept. of CE, GCE Kannur Dr.RajeshKN
50
51. Relation between shear force and bending momentRelation between shear force and bending moment
w
M M dM+
A C
V
V dV+
B D
0Y =â
0BM =â
dx
0Y =â
( ) . 0V V dV w dxâ + â = ( ) ( )
2
0
2
dx
M w V dV dx M dM+ + + â + =
.dV w dxâ =
dVâ
0Vdx dMâ =
dM
V =
Dept. of CE, GCE Kannur Dr.RajeshKN
51
w
dx
= V
dx
=
52. Example 1Example 1
2
x
Px
M =
2
x
x
dM P
V
dx
= = Shear force
0xdV
dx
â
= Load intensity
Dept. of CE, GCE Kannur Dr.RajeshKN
53. Example 2Example 2
w kN/m
Lx L
2
wL
2
wL
2
2 2
x
wLx wx
M = â
2
2
x
x
dM wL
wx V
dx
= â = Shear force
xdV
w
dx
â
= Load intensity
Dept. of CE, GCE Kannur Dr.RajeshKN
dx
54. Example 3Example 3
l
B
l
x
x is taken in the negative direction
2
2
x
wx
M
â
=
x is taken in the negative direction
Hence dx is negative
2
x x
x
dM dM
wx V
dx dx
â
= = = Shear forcedx dxâ
x xdV dV
w
dx dx
â
= = Load intensity
Shear force
Dept. of CE, GCE Kannur Dr.RajeshKN
54
dx dxâ
y
55. Shear force and bending moment diagrams by integrationShear force and bending moment diagrams by integration
dV
w
â
=
dM
V =( ) 1V wdx Câ´ = â +⍠2M Vdx Câ´ = +âŤ
⢠Slope of SFD at any point is the load intensity at that cross section
dx dx
( ) 1⍠2âŤ
Slope of SFD at any point is the load intensity at that cross section
⢠Slope of BMD at any point is the shear force at that cross section
⢠Area of load diagram on the left (or right) of a cross section +g ( g )
reaction = shear force at that cross section
⢠Area of SFD on a segment of a beam = Change in bending moment
at that cross section i.e., dM Vdx=
⢠For bending moment M to be a maximum or minimum,
i.e.0 0,
dM
V= =
Dept. of CE, GCE Kannur Dr.RajeshKN
i.e.0 0, V
dx
56. Example 1
Total load = W
maxw L
W=
max
2
2
W
W
w
L
=
â´ =
L
L
3
W 2
3
W
dVâ
( )âŤ
2w W
To draw SFD
dV
w
dx
= ( ) 1V wdx Câ´ = â +⍠max
2
2
At ,
w W
x w x x
L L
= =
2
1 12 2
2 2
2
W W x
V xdx C C
L L
â â
â´ = + = +âŤ
Dept. of CE, GCE Kannur Dr.RajeshKN
57. 22
12
Wx
V C
L
â
â´ = +
W
W W
At 0,
3
W
x V= =
x
maxw W
1 10
3 3
W W
C Câ´ = + â =
max
2
2
w x x
L L
= =2
2
3
Wx W
V
L
â´
â
+= Parabolic variation
W
3 2
3
Wâ
SFD
Dept. of CE, GCE Kannur Dr.RajeshKN
3SFD
58. To draw BMD
dM
V
dx
=
2
22
3
Wx W
M dx C
L
â ââ
â´ = + +â â
â â
âŤâ â
3
22
i.e.,
3 3
Wx Wx
M C
L
â
= + +
3 3L
At 0 0x M= = 2i.e., 0 C=
3
Wx Wx
M
â
â´ = +At 0, 0x M 2,
2
3 3
M
L
â´ = +
Cubic variation
i.e.0 0,
dM
V= =
For maximum bending moment,
2
9 3
WL
i.e.0 0, V
dx
2
0
Wx W Lâ
+ â
9 3
3L
Dept. of CE, GCE Kannur Dr.RajeshKN
BMD
2
0
3 3
x
L
+ = â =
59. Example 2
d
( ) 1V wdx C= â +âŤ
PP
To draw SFD
âŤ
F t0 0
L
Portion AB
L/4 L/4
A
B C
D
From to0 , 0
4
x x w= = =
V Câ´ =
L
PP
Constant1V Câ´ =
But at 0,x V P= = 1C Pâ = from 0 to,
L
x xV P =â´ ==
Constant
, 1 from 0 to
4
, x xV Pâ´
3L L L
Portion BC
From to
3
, 0
4 4
L L
x x w= = = Also, at , 0
4
L
x V= =
3L L
Dept. of CE, GCE Kannur Dr.RajeshKN
3
from to
4
0,
4
L L
xV x= =â´ = Constant
60. P
PâPâ
SFD
To draw BMD
Portion AB
2M Vdx C= +âŤ
2M Pdx Cⴠ= +⍠2Px C= +From to0 ,
4
L
x x V P= = =
Also, at 0, 0x M= = 2 0Câ = , from 0
4
to
L
M Px x xâ´ = = =
Dept. of CE, GCE Kannur Dr.RajeshKN
Linear variation
61. Portion BC
2M Vdx C= +âŤ
Constant2M Câ´ =From to
3
, 0
4 4
L L
x x V= = =
,
4 4
Also, at
L PL
x M= = 2
4
PL
Câ = from
3
,
4
t
4
o
4
PL L L
M x xâ´ = = =
4 4 4
L L
4
PL
4
PL
BMD4 4
Dept. of CE, GCE Kannur Dr.RajeshKN
62. SummarySummary
Torsion - torsion of circular elastic bars - statically indeterminate
problems - torsion of inelastic circular bars
Axial force, shear force and bending moment - diagrammatic
conventions for supports and loading, axial force, shear force and
bending moment diagrams - shear force and bending moments bybending moment diagrams shear force and bending moments by
integration and by singularity functions
Dept. of CE, GCE Kannur Dr.RajeshKN
62