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Tutorial 4 MECH 101
Liang [email protected]
Office phone : 2358-8811Mobile : 6497-0191
Office hour : 14:00-15:00 Fri
1
Outline
Example about normal stress and normal strain
Example about shear stress, shear strain, and Hooke’s law in shear
Example 1The two bar truss ABC has pin supports at points A and C, which are 2.0m
apart. Members AB and BC are steel bars, pin connected at joint B. The length of BC is 3m. A sign weighing 5.4kn is suspended from bar BC at D and E, which are located 0.8m and 0.4m, respectively, from the ends of the bar. Determine the required cross-sectional area of AB and the required diameter of pin at C if the allowable stresses in tension and shear are 125 MPa and 45 Mpa.
0, 2.7 2.7 0
3.060 .vert AV CV
AV
F R R kN kN
R kN
2 2
2 2
( ) ( ) 5.516
( ) ( ) 5.152
A AV AH
CV CH
R R R kN
Rc R R kN
Return to free body diagram of the entire truss,
Reaction force at A and C,
Supposing counterclockwise moments are positive,
0, (2.0 ) (2.7 )(0.8 ) (2.7 )(2.6 ) 0
4.590
0, 4.590
0, (3.0 ) (2.7 )(2.2 ) (2.7 )(0.4 ) 0
2.340
AH
AH
horiz CH AH
B cv
CV
Mc R m kN m kN m
R kN
F R R kN
M R m kN m kN m
R kN
2
2
5.516
5.252
44.1
57.22
4 / 8.54
AB A
c c
ABAB
allow
cpin
allow
pin pin
F R kN
V R kN
FA mm
VA mm
d A mm
Tensile force in bar ABShear force acting on the pin at C
Required area of the bar
Required diameter of the pin
Example 2 A flexible connection consisting of rubber pads( t=9mm) bonded to steel
plates is shown in the figure. The pads are 160mm long and 80mm wide.
(a)Find the average shear strain γaver in the rubber if the force P=16kN and the shear modulus for the rubber is G=1250kPa.
(B) Find the relative horizontal displacement δ between the interior plate and the outer plates.
Vaver ab
e
aver VabGGe
Shear strain :
In most practical situations the shear strain γ is a small angle, and in such cases tan γ can be replaced by γ.
Shear force V=P/2,Shear Strain:
Horizontal displacement δ,
0.5e
Vaver ab
aver VabGGe
4.5hV
d h mmabGe
Shear stress and Bearing stress
A
Vaver
b
bb A
F PaLFb
Shear stress : tangential to the surface
Average shear stress:
Average bearing stress: Where dLAb
22
PaLV
4
2dA
Where
n
V
m
q
V
p
aL nmp
4
2dA
Example 3 A steel strut S serving as a brace for a boat hoist transmits a compressive forces P=54KN to the deck of a pier. The strut has a hollow square cross section with wall thickness t=12mm, and angle θ is 40 °. A pin through the strut transmits the compressive force from the strut to two gussets G that welded to the base plate B. Four anchor bolts fasten the base plate to deck. The diameter of the pin is dpin=18mm, the thickness of the gussets is tG=15mm, the thickness of the base plate is tB=8mm, the diameter of the anchor bolts is dbolt=12mm. Determine the following stress: (a)the bearing stress between the strut and the pin;(b) the shear stress in the pin;(c) the bearing stress between the pin and the gussets,(d) the bearing stress between the anchor bolts and the base plate,(e) the shear stress in the anchor bolts.
Solution (a)the bearing stress between the strut and the pin
= The force in the strut / the total bearing area of the strut
(b) the shear stress in the pin( the pin tends
to shear the two planes between the gussets and
the strut)
1
54125
2 2(12 )(18 )bpin
P kNMPa
td mm mm
Bearing surface
Shearing surface
(c) the bearing stress between the pin and the gussets
The pin bear against the gussets at two locations, so the bearing area is twice the thickness of the gussets times the pin diameter;
(d) the bearing stress between the anchor bolts and the base plate
(The vertical component of force is transmitted to the pier by direct bearing between the base plate and the pier; the horizontal component, is transmitted through the anchor bolts.)
cos 40 (54 )(cos 40 )108
4 4(8 )(12 )boltB bolt
P kNMPa
t d mm mm
Bearing force
Bearing surface
Bearing surface
(e) the shear stress in the anchor bolts
2 2
cos 40 (54 )(cos 40 )91.4
4 / 4 4 (12 ) / 4boltbolt
P kNMPa
d mm
Shearing force
Shearing surface