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FIRST MULTICOLOUR EDITION
A TEXTBOOK OF
(S.I. UNITS)
[A Textbook for the Students of B.E. / B.Tech.,
U.P.S.C. (Engg. Services); Section ‘B’ of A.M.I.E. (I)]
R.S. KHURMI
J.K. GUPTA
2005
EURASIA PUBLISHING HOUSE (PVT.) LTD.
RAM NAGAR, NEW DELHI-110 055
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Torsional and Bending Stresses in Machine Parts 165
∴ Direct tensile stress,
P P
σo = =
A b1× t
M P.e 6 P.e ⎛ t (b1)2 ⎞
and bending stress, σb = = = ... ⎜Q Z = ⎟
Z Z t (b1) 2 ⎝ 6 ⎠
∴Total stress due to eccentric loading
6 P.e P P ⎛6 e ⎞
= σb + σ o = + = ⎜ + 1⎟
t (b1) 2 b 1× t t . b1 ⎝ b1 ⎠
2
Since the permissible tensile stress is the same as 70 N/mm , therefore
80 × 103 ⎛ 6 × b1 ⎞ 16 × 10
3
⎛ b ⎞
70 = ⎜ + 1⎟ = ... ⎜Q Eccentricity, e = 1 ⎟
20 b1 ⎝ b1 × 2 ⎠ b1 ⎝ 2⎠
∴ 3
b1 = 16 × = 228.6 say 230 mm Ans.
Example 5.23. A cast-iron link, as shown in Fig. 5.25, is to carry a load of 20 kN. If the tensile
and compressive stresses in the link are not to exceed 25 MPa and 80 MPa respectively, obtain the
dimensions of the cross-section of the link at the middle of its length.
Fig. 5.25
Solution. Given : P = 20 kN = 20 × 103 N ; σt(max) = 25 MPa = 25 N/mm2 ; σc(max) = 80 MPa
= 80 N/mm 2
Since the link is subjected to eccentric loading, therefore there
will be direct tensile stress as well as bending stress. The bending
stress at the bottom of the link is tensile and in the upper portion is
compressive.
We know that cross-sectional area of the link,
2a
A = 3a × a + 2 × × 2a
3
= 5.67 a2 mm2
∴ Direct tensile stress,
Fig. 5.26
P 20 × 103 3530
σo = = 2
= 2 N/mm 2
A 5.67 a a
Now let us find the position of centre of gravity (or neutral axis) in order to find the bending
stresses.
Let y = Distance of neutral axis (N.A.) from the bottom of the link as shown
in Fig. 5.26.
a 4a 2
3a 2 × + 2× × 2a
∴ y = 2 3 = 1.2 a mm
5.67 a 2
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268 A Textbook of Machine Design
6. Flanged joint. It is one of the most widely used pipe joint. A flanged joint may be made with
flanges cast integral with the pipes or loose flanges welded or screwed. Fig. 8.8 shows two cast iron
pipes with integral flanges at their ends. The flanges are connected by means of bolts. The flanges
Fig. 8.8. Flanged joint.
have seen standardised for pressures upto
2 N/mm2. The flange faces are machined
to ensure correct alignment of the pipes.
The joint may be made leakproof by
placing a gasket of soft material, rubber
or convass between the flanges. The
flanges are made thicker than the pipe
walls, for strength. The pipes may be
strengthened for high pressure duty by
increasing the thickness of pipe for a short
length from the flange, as shown in Fig. 8.9. Fig. 8.9. Flanged joint.
For even high pressure and for large
diameters, the flanges are further strengthened by ribs or stiffners as shown in Fig. 8.10 (a). The ribs
are placed between the bolt holes.
Fig. 8.10
For larger size pipes, separate loose flanges screwed on the pipes as shown in Fig. 8.10 (b) are
used instead of integral flanges.
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Pipes and Pipe Joints 269
7. Hydraulic pipe joint. This type of joint has oval flanges and are fastened by means of two bolts,
as shown in Fig. 8.11. The oval flanges are usually used for small pipes, upto 175 mm diameter. The
flanges are generally cast integral with the pipe ends. Such joints are used to carry fluid pressure varying
from 5 to 14 N/mm2. Such a high pressure is found in hydraulic applications like riveting, pressing, lifts
etc. The hydraulic machines used in these installations are pumps, accumulators, intensifiers etc.
Fig. 8.11. Hydraulic pipe joint.
8.5 Standard Pipe Flanges for Steam
The Indian boiler regulations (I.B.R.) 1950 (revised 1961) have standardised all dimensions of
pipe and flanges based upon steam pressure. They have been divided into five classes as follows:
Class I : For steam pressures up to 0.35 N/mm2 and water pressures up to 1.4 N/mm2. This is
not suitable for feed pipes and shocks.
Class II : For steam pressures over 0.35 N/mm2 but not exceeding 0.7 N/mm2.
Class III : For steam pressures over 0.7 N/mm2 but not exceeding 1.05 N/mm2.
Class IV : For steam pressures over
1.05 N/mm2 but not exceeding 1.75 N/mm2.
Class V : For steam pressures from
1.75 N/mm2 to 2.45 N/mm2.
According to I.B.R., it is desirable that
for classes II, III, IV and V, the diameter of
flanges, diameter of bolt circles and number
of bolts should be identical and that
difference should consist in variations of the
thickness of flanges and diameter of bolts
only. The I.B.R. also recommends that all
nuts should be chamfered on the side bearing
on the flange and that the bearing surfaces
The Trans-Alaska Pipeline was built to carry oil across the
of the flanges, heads and nuts should be true. frozen sub-Arctic landscape of North America.
The number of bolts in all cases should be a
multiple of four. The I.B.R. recommends that for 12.5 mm and 15 mm bolts, the bolt holes should be
1.5 mm larger and for higher sizes of bolts, the bolt holes should be 3 mm larger. All dimensions for
pipe flanges having internal diameters 1.25 mm to 600 mm are standardised for the above mentioned
classes (I to V). The flanged tees, bends are also standardised.
Visit : www.Civildatas.com
FIRST MULTICOLOUR EDITION
A TEXTBOOK OF
(S.I. UNITS)
[A Textbook for the Students of B.E. / B.Tech.,
U.P.S.C. (Engg. Services); Section ‘B’ of A.M.I.E. (I)]
R.S. KHURMI
J.K. GUPTA
2005
EURASIA PUBLISHING HOUSE (PVT.) LTD.
RAM NAGAR, NEW DELHI-110 055
Visit : www.Civildatas.com
, Visit : www.Civildatas.com
Torsional and Bending Stresses in Machine Parts 165
∴ Direct tensile stress,
P P
σo = =
A b1× t
M P.e 6 P.e ⎛ t (b1)2 ⎞
and bending stress, σb = = = ... ⎜Q Z = ⎟
Z Z t (b1) 2 ⎝ 6 ⎠
∴Total stress due to eccentric loading
6 P.e P P ⎛6 e ⎞
= σb + σ o = + = ⎜ + 1⎟
t (b1) 2 b 1× t t . b1 ⎝ b1 ⎠
2
Since the permissible tensile stress is the same as 70 N/mm , therefore
80 × 103 ⎛ 6 × b1 ⎞ 16 × 10
3
⎛ b ⎞
70 = ⎜ + 1⎟ = ... ⎜Q Eccentricity, e = 1 ⎟
20 b1 ⎝ b1 × 2 ⎠ b1 ⎝ 2⎠
∴ 3
b1 = 16 × = 228.6 say 230 mm Ans.
Example 5.23. A cast-iron link, as shown in Fig. 5.25, is to carry a load of 20 kN. If the tensile
and compressive stresses in the link are not to exceed 25 MPa and 80 MPa respectively, obtain the
dimensions of the cross-section of the link at the middle of its length.
Fig. 5.25
Solution. Given : P = 20 kN = 20 × 103 N ; σt(max) = 25 MPa = 25 N/mm2 ; σc(max) = 80 MPa
= 80 N/mm 2
Since the link is subjected to eccentric loading, therefore there
will be direct tensile stress as well as bending stress. The bending
stress at the bottom of the link is tensile and in the upper portion is
compressive.
We know that cross-sectional area of the link,
2a
A = 3a × a + 2 × × 2a
3
= 5.67 a2 mm2
∴ Direct tensile stress,
Fig. 5.26
P 20 × 103 3530
σo = = 2
= 2 N/mm 2
A 5.67 a a
Now let us find the position of centre of gravity (or neutral axis) in order to find the bending
stresses.
Let y = Distance of neutral axis (N.A.) from the bottom of the link as shown
in Fig. 5.26.
a 4a 2
3a 2 × + 2× × 2a
∴ y = 2 3 = 1.2 a mm
5.67 a 2
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, Visit : www.Civildatas.com
268 A Textbook of Machine Design
6. Flanged joint. It is one of the most widely used pipe joint. A flanged joint may be made with
flanges cast integral with the pipes or loose flanges welded or screwed. Fig. 8.8 shows two cast iron
pipes with integral flanges at their ends. The flanges are connected by means of bolts. The flanges
Fig. 8.8. Flanged joint.
have seen standardised for pressures upto
2 N/mm2. The flange faces are machined
to ensure correct alignment of the pipes.
The joint may be made leakproof by
placing a gasket of soft material, rubber
or convass between the flanges. The
flanges are made thicker than the pipe
walls, for strength. The pipes may be
strengthened for high pressure duty by
increasing the thickness of pipe for a short
length from the flange, as shown in Fig. 8.9. Fig. 8.9. Flanged joint.
For even high pressure and for large
diameters, the flanges are further strengthened by ribs or stiffners as shown in Fig. 8.10 (a). The ribs
are placed between the bolt holes.
Fig. 8.10
For larger size pipes, separate loose flanges screwed on the pipes as shown in Fig. 8.10 (b) are
used instead of integral flanges.
Visit : www.Civildatas.com
, Visit : www.Civildatas.com
Pipes and Pipe Joints 269
7. Hydraulic pipe joint. This type of joint has oval flanges and are fastened by means of two bolts,
as shown in Fig. 8.11. The oval flanges are usually used for small pipes, upto 175 mm diameter. The
flanges are generally cast integral with the pipe ends. Such joints are used to carry fluid pressure varying
from 5 to 14 N/mm2. Such a high pressure is found in hydraulic applications like riveting, pressing, lifts
etc. The hydraulic machines used in these installations are pumps, accumulators, intensifiers etc.
Fig. 8.11. Hydraulic pipe joint.
8.5 Standard Pipe Flanges for Steam
The Indian boiler regulations (I.B.R.) 1950 (revised 1961) have standardised all dimensions of
pipe and flanges based upon steam pressure. They have been divided into five classes as follows:
Class I : For steam pressures up to 0.35 N/mm2 and water pressures up to 1.4 N/mm2. This is
not suitable for feed pipes and shocks.
Class II : For steam pressures over 0.35 N/mm2 but not exceeding 0.7 N/mm2.
Class III : For steam pressures over 0.7 N/mm2 but not exceeding 1.05 N/mm2.
Class IV : For steam pressures over
1.05 N/mm2 but not exceeding 1.75 N/mm2.
Class V : For steam pressures from
1.75 N/mm2 to 2.45 N/mm2.
According to I.B.R., it is desirable that
for classes II, III, IV and V, the diameter of
flanges, diameter of bolt circles and number
of bolts should be identical and that
difference should consist in variations of the
thickness of flanges and diameter of bolts
only. The I.B.R. also recommends that all
nuts should be chamfered on the side bearing
on the flange and that the bearing surfaces
The Trans-Alaska Pipeline was built to carry oil across the
of the flanges, heads and nuts should be true. frozen sub-Arctic landscape of North America.
The number of bolts in all cases should be a
multiple of four. The I.B.R. recommends that for 12.5 mm and 15 mm bolts, the bolt holes should be
1.5 mm larger and for higher sizes of bolts, the bolt holes should be 3 mm larger. All dimensions for
pipe flanges having internal diameters 1.25 mm to 600 mm are standardised for the above mentioned
classes (I to V). The flanged tees, bends are also standardised.
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