DESIGN OF
MACHINE ELEMENTS
LECTURE (10)
Prof. DR. / ABDEL SALAM HEMAID
MECHANICAL ENG. DEPT.
DESIGN & PRODUCTION
SHOUBRA FACULTY OF ENG.
BANHA UNIV.
LECTURE (09)
THREADED
JOINTS-CON.
ECCENTRIC LOAD PERPENDICULAR
TO AXIS OF BOLT
• In this analysis, the following assumptions are made:
• (i) The bracket and the steel structure are rigid.
• (ii) The bolts are fitted in reamed and ground holes.
• (iii) The bolts are not preloaded and there are no tensile
stresses due to initial tightening.
• (iv) The stress concentration in threads is neglected.
• (v) All bolts are identical.
ECCENTRIC LOAD PERPENDICULAR
TO AXIS OF BOLT
ECCENTRIC LOAD PERPENDICULAR
TO AXIS OF BOLT
The moment (P x e) tends to tilt the bracket about the edge C. As
shown in Fig. (b), each bolt is stretched by an amount (d) which is
proportional to its vertical distance from the point C.
ECCENTRIC LOAD PERPENDICULAR
TO AXIS OF BOLT
ECCENTRIC LOAD PERPENDICULAR
TO AXIS OF BOLT
EXAMPLE 7.5
• The following data is given for the bracket illustrated in Fig.(a).
P = 25 kN e = 100 mm l1 = 150 mm l2 = 25 mm
• There is no pre-load in the bolts. The bolts are made of plain
carbon steel 45C8 (Syt = 380 N/mm2) and the factor of safety
is 2.5. Using the maximum shear stress theory, specify the size
of the bolts.
Example 7.6 A wall bracket is
attached to the wall by means of
four identical bolts, two at A and
two at B, as shown in Fig. 7.21.
Assuming that the bracket is held
against the wall and prevented
from tipping about the point C
by all four bolts and using an
allowable tensile stress in the
bolts as 35 N/mm2, determine the
size of the bolts on the basis of
maximum principal stress theory.
EXAMPLE 7.7
• A bracket is fastened to the steel
structure by means of six identical
bolts as shown in Fig. (a). Assume
the following data:
• l1 = 300 mm
• l2 = 200 mm
• l3 = 100 mm
• l = 250 mm
• P = 50 kN
• Neglecting shear stress, determine
the size of the bolts, if the maximum
permissible tensile stress in any bolt
is limited to 100 N/mm2.
SOLUTION
• Given P = 50 kN
l = 250 mm
(σ1)max. = 100 N/mm2
• Step I Maximum tensile force
• The force P tends to tilt the bracket about edge C.
• 𝛿 1 ∝ l1
𝛿 2 ∝ l2
𝛿 3 ∝ l3
• force ∝ stress because (P = σA)
• stress ∝ strain because (σ = E 𝜖 )
• strain ∝ stretch because (𝜖 = 𝛿 /l)
• P1 = Cl1
P2 = Cl2
• Pl = 2P1 l1 + 2P2 l2 + 2P3 l3
P3 = Cl3
EXAMPLE 7.8
• A cast iron bracket fixed to the
steel structure is shown in Fig.
(a). It supports a load P of 25
kN. There are two bolts at A
and two bolts at B. The
distances are as follows,
• l1 = 50 mm l2 = 200 mm l = 400
mm
• Determine the size of the bolts,
if maximum permissible tensile
stress in the bolt is 50 N/mm2.
SOLUTION
• Given
P = 25 kN
l = 400 mm
(σt)max. = 50 N/mm2
• The bolts are subjected to following stresses:
• (i) Direct tensile stress due to load P.
• (ii) Tensile stress due to tendency of the bracket to tilt in clockwise
direction about the edge C.
• Step I Direct tensile force Since the bolts are identical, the direct
tensile force on each bolt is given by,
SOLUTION
Step II Tensile force due to tendency of bracket to tilt
The following assumptions are made:
(i) All bolts are identical.
(ii) The bracket and the structure are rigid.
(iii) The bolts are not preloaded and there is no initial tensile stress due to
tightening of the bolt.
• (iv) As shown in Fig. (b), when the load tends to tilt the bracket about the
edge C, each bolt is stretched by an amount (d), which is proportional to
its distance from the tilting edge. Or,
𝛿 1 ∝ l1
𝛿 2 ∝ l2
P1” = C l1
P2” = C2 I2
•
•
•
•
•
From Table 7.1, the standard size of the bolts is M36 (A = 817 mm2).
7.13 ECCENTRIC LOAD ON CIRCULAR BASE
• The following assumptions are made:
• (i) All bolts are identical.
• (ii) The bearing and the structure are rigid.
• (iii) The bolts are not preloaded and there is no tensile stress
due to initial tightening.
• (iv) The stress concentration in the threads is neglected.
• (v) The bolts are relieved of shear stresses by using dowel
pins.
If P1 P2 … are the resisting forces induced in the bolts,
P1 ∝ l1
or,
P1 = Cl1
P2 = Cl2
P3 = Cl3
P4 = Cl4
where C is the constant of proportionality.
(a)
Four bolts are considered in the above analysis. If the procedure is
repeated for n equally spaced bolts, we get the general expression in
the following form:
The force P1 has maximum value when the term (cos a) has minimum
value. The minimum value of (cos α) is (–1), when (α = 180°). With
reference to Fig. 7.26 (b), the bolt 1 will occupy the topmost position,
at the farthest distance form C, when
α = 180°, Substituting α = 180°
• For a general case with n as number of bolts,
EXAMPLE 7.11
• A round flange bearing, as shown in Fig. 7.26(b), is
fastened to the machine frame by means of four cap
screws spaced equally on a 300 mm pitch circle
diameter. The diameter of the flange is 400 mm. The
external force P is 25 kN, which is located at a
distance of 150 mm from the machine frame. There
are two dowel pins to take shear load. The cap
screws are relieved of all shear force. Determine the
size of the cap screws, if the maximum permissible
tensile stress in the cap screw is limited to 50
2
N/mm .
SOLUTION
• Given 2a = 400 mm 2b = 300 mm P = 25 kN
• l = 150 mm (σt)max. = 50 N/mm2
• Step I Maximum force on cap screw
• It is assumed that the direction of the external force P is fixed and
cap screws are located in such a way that two of them are equally
stressed. From Eq. (7.18),
EXAMPLE 7.13
• Figure shows the bracket used in a jib crane to connect the
tie rod. The maximum force in the tie rod is 5 kN, which is
inclined at an angle of 30° with the horizontal. The bracket is
fastened by means of four identical bolts, two at A and two at
B. The bolts are made of plain carbon steel 30C8 (Syt = 400
N/mm2) and the factor of safety is 5. Assume maximum shear
stress theory and determine the size of the bolts.
7.14 TORQUE REQUIREMENT FOR BOLT
TIGHTENING
• A bolted assembly is tightened by applying force to the
wrench handle and rotating the hexagonal nut.
• In certain applications, as in case of the gasketed joint
between the cylinder and the cylinder head of the engine,
the bolts are tightened with a specific magnitude of pre-load
Pi. It is necessary to determine the magnitude of the torque
which will induce this pre-tension. The torque required to
tighten the bolt consists of the following two factors:
• (i) torque required to overcome thread friction and induce
the pre-load, i.e., (Mt); and
• (ii) torque required to overcome collar friction between the
nut and the washer (Mt)c.
• The torque required to overcome thread friction is
given by,