Solved Problems

Problem: A solid structural steel rod of length $2.00\text{ m}$ and constant cross-sectional area $500\text{ mm}^2$ is subjected to a pure axial tensile load of $50.0\text{ kN}$. Determine the total elongation of the rod. Assume the Modulus of Elasticity is $E=200\text{ GPa}$.

Problem: A stepped steel bar ($E = 200\text{ GPa}$) consists of two segments. Segment AB has an area $A_1 = 500\text{ mm}^2$ and a length $L_1 = 1.00\text{ m}$. Segment BC has an area $A_2 = 200\text{ mm}^2$ and a length $L_2 = 1.50\text{ m}$. An axial tensile load of $40.0\text{ kN}$ is applied at the free end C, and the bar is fixed at A. Determine the total elongation.

Problem: An aluminum stepped bar consists of two segments. The top segment (Segment 1) has a length of $1.50\text{ m}$ and an area of $600\text{ mm}^2$. The bottom segment (Segment 2) has a length of $1.00\text{ m}$ and an area of $400\text{ mm}^2$. The bar is suspended from the ceiling. It carries a downward point load of $30.0\text{ kN}$ at the junction between the segments, and a downward point load of $20.0\text{ kN}$ at the free bottom end. Find the total elongation ($E=70.0\text{ GPa}$).

Problem: A vertical heavy steel cable of length $50.0\text{ m}$ and cross-sectional area $100\text{ mm}^2$ hangs freely from a tower. The unit weight of steel is $\gamma=78.5\text{ kN}/\text{m}^3$. Calculate the elongation strictly due to its own weight. Assume $E=200\text{ GPa}$.

Problem: A solid circular steel bar ($E=200\text{ GPa}$) of length $L=2.00\text{ m}$ tapers uniformly from a diameter $d_1 = 100\text{ mm}$ at one end to $d_2 = 50.0\text{ mm}$ at the other end. An axial tensile load of $P=150\text{ kN}$ is applied to both ends. Determine the total elongation of the bar.

Problem: A stepped bar is fixed rigidly at both ends. It has an upper section of length $L_1 = 400\text{ mm}$ and area $A_1 = 600\text{ mm}^2$, and a lower section of length $L_2 = 300\text{ mm}$ and area $A_2 = 300\text{ mm}^2$. A downward load of $P = 50.0\text{ kN}$ is applied at the step. Determine the reactions at the fixed supports. Assume $E = 200\text{ GPa}$ for the entire bar.

Problem: A uniform steel bar is fixed at both ends (A and B). The total length is $3.00\text{ m}$. A point load of $60.0\text{ kN}$ is applied axially at point C, $1.00\text{ m}$ from A (and $2.00\text{ m}$ from B). The area is constant at $300\text{ mm}^2$, and $E = 200\text{ GPa}$. Determine the reactions at A and B.

Problem: A rigid horizontal bar AB of length $3.00\text{ m}$ is hinged at A and supported by a steel wire at B ($L=2.00\text{ m}$, $A=200\text{ mm}^2$, $E=200\text{ GPa}$). A vertical downward load $P=10.0\text{ kN}$ is applied at C, $2.00\text{ m}$ from A. Determine the vertical displacement of point B.

Problem: A continuous brass pipeline is exactly $3.00\text{ m}$ long at an ambient temperature of $20.0^\circ\text{C}$. Find its new length if hot fluid causes the temperature to rise to $80.0^\circ\text{C}$. The coefficient of linear thermal expansion for brass is $\alpha=19.0 \times 10^{-6} /^\circ\text{C}$.

Problem: A steel railroad rail is $10.0\text{ m}$ long and is laid securely between two massive concrete abutments at $15.0^\circ\text{C}$ with absolutely zero gap. Determine the internal compressive stress in the rails if the summer temperature rises to $50.0^\circ\text{C}$ and the extreme pressure causes the concrete abutments to yield (get pushed outward) by exactly $1.00\text{ mm}$. ($E=200\text{ GPa}$, $\alpha=11.7 \times 10^{-6} /^\circ\text{C}$)

Problem: A compound bar consists of a copper tube surrounding a steel core. They are rigidly fastened together at both ends. The initial temperature is $20.0^\circ\text{C}$. The assembly is heated to $80.0^\circ\text{C}$. The copper tube has an area of $1000\text{ mm}^2$, $E = 100\text{ GPa}$, and $\alpha = 17.0 \times 10^{-6} /^\circ\text{C}$. The steel core has an area of $500\text{ mm}^2$, $E = 200\text{ GPa}$, and $\alpha = 12.0 \times 10^{-6} /^\circ\text{C}$. Determine the internal stresses in both materials.