sm9 103

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PROBLEM (9.103) A plate of thickness t = 5 mm is made of a cast-aluminum alloy for which u
σ
= 70 MPa and '
u
σ
= 175 MPa. Calculate the force P required to punch a d = 15 mm-diameter hole
through the plate, as depicted in Fig. P9.103 and discussed in Sec. 2.5. Apply:
(a) The maximum normal stress theory of failure.
(b) Colulomb-Mohr theory of failure.
Assumption: The shear force is taken to be uniformly distributed through the plate thickness.
SOLUTION
Uniform shear stress
τ
acts on a typical element shown in Fig. (a).
(a) 1u
σ
σ = or 3u
σ
σ =
u
P
td
σ
π
=
6
(0.005)(0.015)(70 10 )
u
Ptd πσ π == ×
or
16.49PkN=
(b) 3
11; (1 )
''
uu
uu u
σ
σ
σ
τ
σ
σσ σ
−= + =
From which
6
(0.005)(0.015)(70 10 )
' 175
1
170
u
u
u
td
P
σ
σ
σ
×
==
+
+
or
1.5PkN=
Pdt
τ
π
=
13
σ
στ =− =