sm8 123

_______________________________________________________________________
PROBLEM (*8.123) At a point A on the surface of a vessel made of structural steel, the strains
'
x
ε
and 'y
ε
are measured in the x’ and y’ directions oriented at an angle
θ
to the x and y axes,
respectively (Fig. P8.123). Calculate:
(a) The components of strain
x
ε
, y
ε
, and ''
x
y
γ
.
(b) Poisson’s ratio
ν
for the steel.
Given: '
x
ε
= 250
μ
, 'y
ε
= 400
μ
,
x
y
γ
= 0,
θ
= 35o
Assumption: The stresses remain below yield strength of the material.
*SOLUTION
(a) '' ''
( )sin( 2 ) cos( 2 )
oo
xy x y x y
γ
εε θ γ θ =− − − + −
,
''
0 (250 400)sin( 70 ) cos( 70 )
oo
xy
γ =− − − + − '' 412.1
xy
γ
μ
=
Then
'' '' ''
11 1
()()cos(2) sin(2
22 2
oo
xxy xy xy )
ε
εε εε θ γ θ =++− −+ −
11 1
(250 400) (250 400)cos( 70 ) (412.1)sin( 70 )
22 2
oo
=++− −+ −
105.8
μ
=
'' '' ''
11 1
( ) ( )cos( 2 ) sin( 2 )
22 2
oo
yxy xy xy
ε
εε εε θ γ θ =+−− −− −
11 1
(250 400) (250 400)cos( 70 ) (412.1)sin( 70 )
22 2
oo
=+−− −− −
544.3
μ
=
(b) Hooke’s Law, with , ,
tyax
ε
εε ε == and 2
ta
σ
σ
=
(1 2 ) (2 )
xx
xy
EE
σ
σ
ε
νε =− = − ν
So 544.3 2
5.145
105.8 1 2
y
ε
ν
−
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x
ε
ν
===
−, 0.34
ν
=