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

552
STRAIN TRANSFORMATIONS
13.6 mohr’s circle for plane Strain
The general strain transformation equations, expressed in terms of double-angle trigonometric
functions, were presented in Section 13.3. Equation (13.4) is
ε
n
εx + εy εx − εy γ xy
= + cos2θ
+
2 2
2 sin2 θ
and Equation (13.6) is
γ nt ε x − ε y γ
=− sin2θ
+
xy
2 2
2 cos2 θ
Equation (13.4) can be rewritten so that only terms involving 2θ appear on the right-hand side:
εx + εy εx − εy γ xy
εn
− = cos2θ
+
2 2
2 sin2 θ
γ nt
εx − εy γ xy
=− sin2θ
+
2 2
2 cos2 θ
Both equations can be squared, then added together, and simplified to give
⎛
εn
⎝
⎜
2 2 2 2
εx + εy⎞
γ nt
εx εy γ xy
−
2 ⎠
⎟ + ⎛ ⎞ ⎛ − ⎞
=
⎝ 2 ⎠ ⎝
⎜
2 ⎠
⎟ + ⎛ ⎝ ⎜ ⎞
2 ⎠
⎟ (13.13)
This is the equation of a circle in terms of the variables ε n and γ nt /2. It is similar in form
to Equation (12.20), which was the basis of Mohr’s circle for stress.
Mohr’s circle for plane strain is constructed and used in much the same way as
Mohr’s circle for plane stress. The horizontal axis used in the construction is the ε axis,
and the vertical axis is γ /2. The circle is centered at
on the ε axis, and it has a radius
εx
+ εy
C =
2
R =
2 2
⎛ εx − εy⎞
γ xy
⎝
⎜
2 ⎠
⎟ + ⎛ ⎝ ⎜ ⎞
2 ⎠
⎟
There are two notable differences in constructing and using Mohr’s circle for strain, compared
with Mohr’s circle for stress. First, note that the vertical axis for the strain circle is γ /2;
hence, shear strain values must be divided by 2 before they are plotted. Second, the endpoints
of the diameter have the coordinates (ε x , − γ xy /2) and (ε y , γ xy /2).
ExAmpLE 13.3
y
O
π
2
– γ xy
x
The strain components at a point in a body subjected to plane strain are ε x = 435 µε,
ε y = −135 µε, and γ xy = −642 µrad. The deflected shape of an element that is subjected to
these strains is shown. Determine the principal strains, the maximum in-plane shear
strain, and the absolute maximum shear strain at point O. Show the principal strain deformations
and the maximum in-plane shear strain distortion in a sketch.