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

38
STRAIN
In this expression, θ′ is the angle in the deformed state between two initially orthogonal
reference lines.
y
π
– γ
2 xy
x
FIGURE 2.5a A positive value
for the shear strain γ xy means that
the angle θ′ between the x and y
axes decreases in the deformed
object.
Units of Strain
Equations (2.3) through (2.5) indicate that shear strains are dimensionless angular quantities,
expressed in radians (rad) or microradians (µrad). The conversion from radians, a
dimensionless quantity, to microradians is 1 µrad = 1 × 10 −6 rad.
measuring Shear Strains Experimentally
Shear strain is an angular measure, and it is not possible to directly measure the extremely
small angular changes typical of engineered structures. However, shear strain can be determined
experimentally by using an array of three strain gages called a strain rosette. Strain
rosettes will be discussed in more detail in Chapter 13.
y
π
– γ
2 xy
x
Sign conventions for Shear Strains
Equation (2.5) shows that shear strains will be positive if the angle θ ′ between the x and y
axes decreases. If the angle θ′ increases, the shear strain is negative. To state this relationship
another way, Equation (2.5) can be rearranged to give the angle θ′ in the deformed
state between two reference lines that are initially 90° apart:
π
θ ′ = − γ
2
xy
FIGURE 2.5b The angle
between the x and y axes
increases when the shear
strain γ xy has a negative value.
If the value of γ xy is positive, then the angle θ′ in the deformed state will be less than 90°
(i.e., less than π /2rad) (Figure 2.5a). If the value of γ xy is negative, then the angle θ′ in the
deformed state will be greater than 90° (Figure 2.5b). Positive and negative shear strains
are not given special or distinctive names.
ExAmpLE 2.2
y
R
12 in.
P
8 in.
S
V
Q
x
0.0625 in.
The shear force V shown causes side QS of the thin rectangular plate to displace
downward 0.0625 in. Determine the shear strain γ xy at P.
Plan the Solution
Shear strain is an angular measure. Determine the angle between the x axis and side
PQ of the deformed plate.
SOLUTION
Determine the angles created by the 0.0625 in. deformation. Note: The small-angle
approximation will be used here; therefore, sin γ ≈ tan γ ≈ γ , and we have
0.0625 in.
γ = = 0.0078125 rad
8in.