Question 2.5: When the “rigid” beam AB in Fig. 1 is horizontal, the rod BC......
When the “rigid” beam AB in Fig. 1 is horizontal, the rod BC is strain free. (a) Determine an expression for the average extensional strain in rod BC as a function of the angle θ of clockwise rotation of AB in the range 0 ≤ θ ≤ π/2. (b) Determine an approximation for ε(θ) that gives acceptable accuracy for values of ε when θ \ll 1 rad.
Plan the Solution The defining equation for extensional strain, Eq. 2.6, can be used to determine the required expression for the average extensional strain of rod BC. To determine the geometrical relationship between the extended length L* and the angle θ , we can draw a sketch of the deformed rod-beam system (\text{i.e., a deformation diagram}).^5 Since beam AB is assumed to be rigid, end B moves in a circle about end A.
(a) Obtain an expression for ε(θ). From Eq. 2.6,
\epsilon_{avg} = \frac{ΔL}{L}=\frac{L^* – L}{L} Average Extensional Strain (2.6)
\epsilon =\frac{L^* – L}{L}From the original figure,
L ≡ \overline{BC} = 5aDeformation Diagram: Rigid beam AB rotates about A, while rod BC stretches and rotates about C, as indicated in Fig. 2.
Geometry of Deformation: Using the Pythagorean theorem and dimensions from the sketch of the deformed configuration in Fig. 2, we get
L^* = \sqrt{(3a + c^*)^2 + (b^*)^2}where
b* = 4a cos θ, c* = 4a sin θ
Then, from Eq. 2.6,
\epsilon =\frac{L^* – L}{L} = \frac{a\sqrt{(3+4 \sin θ)^2 + (4 \cos θ)^2} – 5a}{5a}This can be simplified to
ε(θ) = \sqrt{1+ \left(\frac{24}{25}\right)\sin θ} – 1 (1)
(b) Approximate ε(θ) for θ \ll 1 rad.^6 If θ \ll 1 , then sin θ ≈ θ.
Furthermore, the term under the radical has the form 1 + β, with β \ll 1 .
According to the binomial expansion theorem, for small values of β
Thus, for θ \ll 1 ,ε(θ) takes the form
ε(θ) = \left(\frac{24}{25}\right) θ, θ \ll 1 rad (2)
Review the Solution In both answers, ε(θ) is dimensionless, as it should
be. If Eq. (1) is evaluated at θ =π/2, we get ε(π/2) = 2/5. Since L* = 3a
+ 4a = 7a when B* is directly below A, this value of ε(π/2) is correct.
