Deformation of Tapered Bars

Q. A mild steel rod tapers uniformly from $30mm$ diameter to $12mm$ diameter in a length of $300mm$. The rod is subjected to an axial load of $12kN$. $E=2\times {10}^{5}N/m{m}^2$. What is the extension of the rod in mm?

1. $\frac{4\pi }{5}$
2. $\frac{4}{5\pi }$
3. $\frac{\pi }{5}$
4. $\frac{1}{5\pi }$

Q. A bar of circular cross-section varies uniformly from a cross-section 2D to D. If extension of the bar is calculated treating it as a bar of average diameter, then the percentage error will be

1. 10
2. 25
3. 33.33
4. 50

Q. A tapered circular rod of diameter varying from $20mm$ to $10mm$ is connected to another uniform circular rod of diameter $10mm$ as shown in the following figure. Both bars are made of same material with the modulus of elasticity, $E=2\times {10}^5$MPa. When subjected to a load $P=30\pi kN$, the deflection at point $A$ is

1. 15

Q. A round bar made of same material consists of 3 parts each of 100 mm length having diameters of 40mm, 50 mm and 60 mm, respectively. If the bar is subjected to an axial load of 10 kN, the total elongation of the bar in mm would be (E is the modulus of elasticity in kN/mm${}^2$)

1. $\frac{0.4}{\pi E}(\frac{1}{16}+\frac{1}{25}+\frac{1}{36})$
2. $\frac{4}{\pi E}(\frac{1}{16}+\frac{1}{25}+\frac{1}{36})$
3. $\frac{4\sqrt{2}}{\pi E}(\frac{1}{16}+\frac{1}{25}+\frac{1}{36})$
4. $\frac{4D}{\pi E}(\frac{1}{16}+\frac{1}{25}+\frac{1}{36})$

Q. A mild steel bar, circular in cross-section, tapers from $40mm$ diameter to $20mm$ diameter over its length of $800mm$. It is subjected to an axial pull of $20kN$. $E=2\times {10}^{5}N/m{m}^2$. The increase in the length of the rod will be

1. $\frac{1}{10\pi }mm$
2. $\frac{2}{5\pi }mm$
3. $\frac{4}{5\pi }mm$
4. $\frac{1}{5\pi }mm$

Q. Assertion (A): A bar tapers from a diameter of ${}^{\prime }{d}_1^{\prime }$ to a diameter of '$d_2^{\prime }$ over its length $L$ and is subjected to a tensile force $P$. If extension is calculated based on treating it as a bar of average diameter, the calculated extension will be more than the actual extension.
Reason (R): The actual extension in such bars is given by, $\Delta =\frac{4PL}{\pi d_1{d}_{2}E}$

1. both A and R are true and R is the correct explanation of A
2. both A and R are true but R is not a correct explanation of A
3. A is true but R is false
4. A is false but R is true

Q. A mild steel bar of length $450mm$ tapers uniformly. The diameters at the ends are $36mm$ and $18mm$, respectively. An axial load of $12kN$ is applied on the bar. $E=2\times {10}^{5}N/m{m}^2$. The elongation of the bar will be

1. $\frac{1}{3\pi }mm$
2. $\frac{1}{6\pi }mm$
3. $\frac{3\pi }{2}mm$
4. $\frac{2}{3\pi }mm$

Q. A horizontal bar, fixed at one end (x=0), has a length of 1 m, and cross-sectional area of 100 $m{m}^2$. Its elastic modulus varies along its length as given by E(x) = 100 $e^{-x}$ GPa, where x is the length coordinate (in m) along the axis of the bar. An axial tensile load of 10kN is applied at the free end (x =1). The axial displacement of the free end is mm.

1. 1.718

Q. A mild steel rod tapers uniformly from $24mm$ dia to $12mm$ dia over its length of $400mm$. The rod when held vertical is subjected to an axial tensile load of $12kN$. $E=2\times {10}^{5}N/m{m}^2.$ The extension of the rod in $mm$ would be

1. $\frac{3\pi }{2}$
2. $\frac{2}{3\pi }$
3. $\frac{\pi }{3}$
4. $\frac{1}{3\pi }$