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

13.4 principal Strains and maximum
Shearing Strain
Given the similarity among Equations (13.3), (13.4), (13.5), and (13.6) for plane strain and
Equations (12.5), (12.6), (12.7), and (12.8) for plane stress, it is not surprising that all of the
relationships developed for plane stress can be applied to plane strain analysis, provided that
the substitutions given in Table 13.1 are made. Expressions for the in-plane principal directions,
the in-plane principal strains, and the maximum in-plane shear strain are as follows:
ε
p1, p2
γ xy
tan2θ
p =
ε − ε
x
y
2 2
(13.9)
εx + εy ⎛ εx − εy⎞
γ xy
= ±
2 ⎝
⎜
2 ⎠
⎟ + ⎛ ⎝ ⎜ ⎞
2 ⎠
⎟ (13.10)
2 2
γ max ⎛ εx − εy⎞
γ xy
=±
2 ⎝
⎜
2 ⎠
⎟ + ⎛ ⎝ ⎜ ⎞
2 ⎠
⎟ (13.11)
Equations (13.9), (13.10), and
(13.11) are similar in form to
Equations (12.11), (12.12), and
(12.15), respectively. However,
instances of τ xy in the stress
equations are replaced by γ xy /2
in the strain equations. Be
careful not to overlook these
factors of 2 when switching
between stress analysis and
strain analysis.
In the preceding equations, normal strains that cause elongation (i.e., a stretching produced
by a tensile stress) are positive. Positive shear strains decrease the angle between the element
faces at the coordinate origin. (See Figure 13.3.)
As was true in plane stress transformations, Equation (13.10) does not indicate which
principal strain, either ε p1 or ε p2 , is associated with the two principal angles. The solution of
Equation (13.9) always gives a value of θ p between −45° and +45° (inclusive). The principal
strain associated with this value of θ p can be determined from the following two-part rule:
• If the term ε x − ε y is positive, then θ p indicates the orientation of ε p1 .
• If the term ε x − ε y is negative, then θ p indicates the orientation of ε p2 .
The other principal strain is oriented perpendicular to θ p .
The two principal strains determined from Equation (13.10) may be both positive, both
negative, or positive and negative. In naming the principal strains, ε p1 is the more positive
value algebraically (i.e., the algebraically larger value). If one or both of the principal strains
from Equation (13.10) are negative, ε p1 can have a smaller absolute value than ε p2 .
Absolute maximum Shear Strain
When a state of plane strain exists, ε x , ε y , and γ xy may have nonzero values. However, strains
in the z direction (i.e., the out-of-plane direction) are zero; thus, ε z = 0 and γ xz = γ yz = 0.
Equation (13.10) gives the two in-plane principal strains, and the third principal strain is
ε p3 = ε z = 0. An examination of Equations (13.10) and (13.11) reveals that the maximum
in-plane shear strain is equal to the difference between the two in-plane principal strains:
γ = ε − ε
(13.12)
max p1 p2
However, the magnitude of the absolute maximum shear strain for a plane strain element
may be larger than the maximum in-plane shear strain, depending upon the relative magnitudes
and signs of the principal strains. The absolute maximum shear strain can be determined
from one of the three conditions shown in Table 13.2.
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