Question 2.1.1: By bending a thin ruler, you are able to deform it into a ci...
By bending a thin ruler, you are able to deform it into a circular arc. This arc, with a radius of 30 in., encloses an angle of 23° at center, as shown. Find the average normal strain developed in the ruler.
Given the initial length of the ruler, L_o , which we assume to be exactly 12 in., and the characteristics of a circular arc formed when it is deformed under bending, we must find the intensity of deformation, or induced strain. Since we know that strain is a measure of the change in a body’s length relative to the original length, we must determine how much the ruler’s length of 12 in. changes under this deformation.
Recalling that the arc length of a circular arc is given by the equation
arc length = rθ
and that in this case, the arc length is the deformed length of the ruler, L, we have
L = r θ = (30 \textrm{ in.}) ⋅ 23° · \frac{2π \textrm{ rad}}{260°}= (30 in.) ⋅ (0.4014 rad) = 12.04277 in.
Normal strain is then calculated
ε =\frac{\textrm{change in length}}{\textrm{original length}}= \frac{L − L_o}{L_o}= \frac{0.04277 \textrm{ in.}}{12 \textrm{ in.}}= 0.003564 \frac{\textrm{in.}}{\textrm{in.}}For convenience, such a small strain might be reported as 3564 micro-inches
per inch (μin./in.), or 3564 microstrain, or alternatively as a 0.36% strain.