sm8 132

_______________________________________________________________________
PROBLEM (8.132) At a point on the free surface of a steel member of modulus of elasticity E and
Poisson’s ratio
ν
, subjected to plane stress, a 60° rosette (Fig. P8.132) measures the strains
a
ε
= 1200
μ
, b
ε
= -650
μ
, c
ε
= 500
μ
Determine:
(a) The principal strains and their directions.
(b) The corresponding principal stresses and the maximum shear stresses.
Sketch the results found in part (b) on a properly oriented deformed element.
SOLUTION
Equation (8.45):
1
1200 [2( 650 500) 1200] 500
3
xy
ε
με ==−+−=− μ
2( 650 500) 1328
3
xy
γ
μ =−− =−
(a) 1
' (1200 500) 350
2
ε
μ =−=
Excerpts from this work may be reproduced by instructors for distribution on a not-for-profit basis for testing or
instructional purposes only to students enrolled in courses for which the textbook has been adopted. Any other
reproduction or translation of this work beyond that permitted by Sections 107 or 108 of the 1976 United States
Copyright Act without the permission of the copyright owner is unlawful.
22 664 850 1079R
μ
=+=
μ
γ
21582
max
=
=
r
1
1 664
'tan[ ]19 19
oo
=− =
2 850
p
θ
−
=−
μ
ε
14291079350
1=+=
μ
ε
7291079350
2
−
=
−
=
(b) Equation (3.16):
3
1200(10 )[1429 0.3( 729)] 266
10.09
M
Pa σ =+−=
−
3
2200(10 )[ 729 0.3(1429)] 66
10.09
M
Pa σ =−+ =−
−
3
max 200(10 ) (2158) 166
2(1 0.3)
M
Pa τ ==
+
Check:
max 266 ( 66) 166
2
M
Pa τ
−−
==
We also have
266 66
'100
2
M
Pa σ
−
==
2
ε
R
ε
(
μ
)
1
ε
2
γ
(μ)
1200, 664)
(
O
C
2'
p
θ
'
ε
x
y
66 MPa
x
’
266 MPa
19o
166 MPa
x
100 MPa