sm8 36 38

_______________________________________________________________________
PROBLEMS (8.36 through 8.38) The stresses at three points in a loaded member are represented
in Figs. P8.36 through P8.38. For each point, calculate and sketch:
(a) The principal stresses.
(b) The maximum shearing stresses with the associated normal stresses.
SOLUTION (8.36)
MPaMPa xyyx 60,0,40
−
=
=−=
τ
σ
σ
(a) 22
1,2 40 40
( ) ( 60)
22
σ
−−
=± +−
24.6320±−=
MPaMPa 2.832.43 21
−
==
σ
σ
12( 60)
2 tan 71.6
40 o
p
θ
−−
==
−
'40 40 cos71.6 60sin71.6
22 oo
x
σ
−−
=+ −
MPa2.839.5631.620
−
=−−−=
Thus,
" 35.8o
p
θ =
(b) max 43.2 83.2 63.2
2
M
Pa τ
+
==
40 0
'20
2
M
Pa σ
−+
==−
63.2 MPa
x
’
83.2 MPa
35.8o
20 MPa
x
43.2 MPa
SOLUTION (8.37)
MPaMPaMPa xyyx 10,20,5
=
−=−=
τ
σ
σ
(a) 22
1,2 520 520
()1
22
σ
+−+
=− ± + 0
= 5.125.12 ±−
MPa250 21 ==
σ
σ
Continued on next slide
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.

12(10)
2 tan 53.1
520 o
p
θ
−
==
−+
'520 520
cos53.1 10sin53.1
22 oo
x
σ
+−+
=− + +
085.45.12 =++−=
x
’
12.5 MPa
26.5o
12.5 MPa
x
25 MPa
Thus, ' 26.5o
p
θ =
(b)max 02512.5
2
M
Pa τ
+
==
520
' 12.5
2
M
Pa σ
−−
==−
SOLUTION (8.38)
ksiksiksi xyyx 7,15,8
−
=
==
τ
σ
σ
(a) 22
1,2 15 8 15 8
( ) ( 7) 11.5 7.826
22
σ
+−
=± +−=±
12
19.33 3.67ksi ksi
σ
σ ==
12( 7)
2 tan 63.4
15 8 o
p
θ
−−
==−
−
'15 8 15 8cos( 63.4 ) 7sin( 63.4 )
22 oo
x
σ
+−
=+ −−−
11.5 1.567 6.259 19.33 ksi=+ + = 11.5 ksi
x
’
19.33 ksi
31.7o
7.83 ksi
x
3.67 ksi
Thus,
' 31.7o
p
θ =−
(b) max 19.33 3.67 7.83
2ksi
τ
−
==
19.33 3.67
' 11.5
2ksi
σ
+
==
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.