Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

The in-plane principal directions can be determined from Equation (13.9):
tan2θ
p
γ xy
=
ε − ε
x
y
−980
980
=
= − −680 − 320 −1000
Note: ε
∴ 2θ
= 44.42° and thus θ = 22.21°
p
p
x
− ε < 0
y
Since ε x − ε y < 0, the angle θ p is the angle between the x direction and the ε p2 direction.
The problem states that this is a plane strain condition. Therefore, the out-of-plane
normal strain ε z = 0 is the third principal strain ε p3 . Since ε p1 is positive and ε p2 is negative,
the absolute maximum shear strain is the maximum in-plane shear strain. Therefore,
the magnitude of the absolute maximum shear strain (see Table 13.2) is
γ = ε − ε = 1, 400 rad
absmax p1 p2 µ
y
520 με
22.21°
880 με
π – 1,400 μrad
2
x
Sketch the Deformations and Distortions
The principal strains are oriented 22.21° counterclockwise from the
x direction. The principal strain corresponding to this direction is ε p2 =
−880 µε; therefore, the element contracts parallel to the 22.21° direction.
In the perpendicular direction, the principal strain is ε p1 = 520 µε, which
causes the element to elongate.
To show the distortion caused by the maximum in-plane shear
strain, connect the midpoints of each of the rectangle’s edges, to create
a diamond. Two interior angles of this diamond will be acute angles
(i.e., less than 90°), and two interior angles will be obtuse (i.e., greater
than 90°). Use the positive value of γ max obtained from Equation (13.11)
to label one of the acute interior angles with π/2 − γ max . The obtuse
interior angles will have a magnitude of π/2 + γ max . Note that the four
interior angles of the diamond (or any quadrilateral) must total 2π radians
(or 360°).
pRoBLEmS
p13.1 The thin rectangular plate shown in Figure P13.1/2 is uniformly
deformed such that ε x = 230 µε, ε y = −480 µε, and γ xy =
−760 µrad. Using dimensions of a = 20 mm and b = 25 mm, determine
the normal strain in the plate in the direction defined by
(a) points O and A.
(b) points O and C.
p13.2 The thin rectangular plate shown in Figure P13.1/2 is
uniformly deformed such that ε x = −360 µε, ε y = 770 µε, and γ xy =
940 µrad. Using dimensions of a = 25 mm and b = 40 mm, determine
the normal strain in the plate in the direction defined by
(a) points O and B.
(b) points O and D.
a
a
a
a
y
O
FIGURE p13.1/2
x
B
D
b b b b
A
b
C
550