Quantitative Problems
2.71 A stress
51
tensor is given by
w"
/zoo 40 o\
':(,';' ': s)
+
304
where all values are in megapascals. Calculate the principal normal stresses and the principal shear stresses.
Ans. ot: 208.7 MPa, oz : 16.29 MPa, a3 : Q.
I
I
(a)
2.72
A thin, square steel plate is oriented with respect to
the z- and g-directions. A tensile stress tr acts in the
160
z-direction, and a compressive stress -a acts in the gdirection. Determine the normal and shear srresses on
thediagonalof thesquare. Ans. oa5.=Q1747'c : -q.
2.73
+
I
I
A thin, rectangular brass plate has normals to the sides
in the r- and E-directions. A tensile stress o acts on the
four sides. Determine the principal normal and shear
stresses. Ans.
ot: oz: o,r=4.
(b)
thin, rectangular brass plate inProbIem2.73,
but with the stress in the 3r-direction being ou : -o instead of ta, determine the principal normal and shear
stresses and their directions. ,4ns. 01 : -o2 : o,
2.74 Given the
--+^
2.78 A certain loading on
2J5 For the following stress states, sketch the stress element,
draw the appropriate Mohr's circle, determine the principal stresses and their directions, and sketch the principal stress elements:
(a) o"
:8,
oi :
6
oo
:
14, andT*a
: 4. Arts. ot:
16 MPa,
MPa.
: -I5, o, : 9, and r,a : 5. Arts. ot : 10 MPa,
: 16 MPa'
(c) o, : 12, o, :28, and r*y : 15. Ans. o1 : 35
MPa, oz: 5 MPa.
(d) o" : -54, oo - 154, and r"o : -153. Atts. o1 :
235 MPa, az : - 135 MPa.
(b) a"
02
All
stresses are
in megapascals.
2.76 Repeat Problem 2.75
:
for
: -I0, and. r*o : 0. Atts. t , : t] ftapi.\
(b) o, - 0, ou :30, and r,a : 20. a,riti.: .fO fr4f'u,
oz: -l0MPa.
(c) o" : -20, oo: 40, and raa - -40. At1s. or : 60
MPa, o2: *40 MPa.
(d) o, : 30, oo : O, and r,o : -20. Ans. ot : 4L)
MPa. oz : -10 MPa.
(a) o,
oy
A1l stresses are
2.77
Sketch ff, used in Problem 2.77
in megapascals.
Given the state of stresses shown in the two parts of
Sketch // determine the principal stresses and iheir directions by using Mohr's circle and the stress equations.
Show the stress elements. All stresses in Sketch f f are
in megapascals. Ans. (a) o1 : 34 MPa, a: : -38 MPa.
state of
or :
Q,
oo
-
a
machine element leads to a stress
af 2, antd
r.o : al4. W|.at value of
a results in the maximum allowable shear stress of 100
MPa? Ans. ri : 282.8 MPa.
2.79 Given the normal and shear stresses or : 4 ksi, a, : 6
ksi, and r,s : -5 ksi, draw the Mohr's circle diagram
and the principal normal and shear stresses on the z-E
axis. Determine the triaxial stresses and give the corresponding Mohr's circle diagram. Ans. ot : 10.10 ksi,
oz : 0, o3 : -0.t)99 ksi.
2.80 Given the normal and shear stresses o* : O, ay : 10 ksi,
and r,o : 12 ksi, draw the Mohr's circle diagram and
the principal normal and shear stresses on the ,r-y axis.
Determine the triaxial stresses and give the corresponding Mohr's circle diagram. Ans. ot : 18 ksi, oz : O,
o-r : -8 ksi'
2.81 Given the normal and shear stresses or : 70ksi, oo 24 ksi, and rra : -12 ksi, draw the Mohr's circle diagram and the principal normal and shear stresses on the
r-y axis. Determine the triaxial stresses and give the corresponding Mohr's circle diagram. Ans. o 1 : 30.89 ksi,
oz
2.82
:
3.1.07
ksi, a3
:
6.
stress element in plane stress encounters o, : 29
MPa, oo :
10 MPa, an:.d r,o : 13 MPa. (a) Determine
the three principal stresses and maximum shear stress.
(b) Using a Mohr's circle diagram, explain the effect of
superimposing a hydrostatic pressure p on the principal
stresses and maximum shear stress. Ans. (a) ot : 2i1.85
MPa, oz : 0, cts: -1ul.B5 MPa.
A
2.83 In a three-dimensional stress field, the stresses are found
to be o" : 4 ksi, oa : 2ksi, o" : 6ksi, r,o : -2 ksi,
ro. : 0, artd T". : 2 ksi. Draw the stress element for
this case. Determine the principal stresses and sketch
the corresponding Mohr's circle diagram. Ans.
7.064ksr, oz : 4.691ksi, o3 : 0.2412 ksi.
ot :