Problem: A stepped steel bar ($E = 200$ GPa) consists of two segments. Segment AB has area $A_1 = 500$ mm$^2$ and length $L_1 = 1$ m. Segment BC has area $A_2 = 200$ mm$^2$ and length $L_2 = 1.5$ m. An axial tensile load of 40 kN is applied at the free end C, and the bar is fixed at A. Determine the total elongation.

Problem: A steel bar is fixed at both ends (A and B). The total length is 3 m. A load of 60 kN is applied axially at point C, 1 m from A (and 2 m from B). The area is constant at 300 mm$^2$, and $E = 200$ GPa. Determine the reactions at A and B.

Problem: A rigid horizontal bar AB of length 3 m is hinged at A and supported by a steel wire at B ($L=2$ m, $A=200$ mm$^2$, $E=200$ GPa). A load $P=10$ kN is applied at C, 2 m from A. Determine the vertical displacement of point B.

Problem: A stepped bar is fixed 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 $P = 50 \text{ kN}$ is applied at the step. Determine the reactions at the fixed supports. Assume $E = 200 \text{ GPa}$.

Problem: A solid structural steel rod of length $2\text{ m}$ and constant cross-sectional area $500\text{ mm}^2$ is subjected to a pure axial tensile load of $50\text{ kN}$. Determine the total elongation of the rod. Assume the Modulus of Elasticity is $E=200\text{ GPa}$.

Problem: An aluminum stepped bar consists of two segments. The top segment (Segment 1) has a length of $1.5\text{ m}$ and an area of $600\text{ mm}^2$. The bottom segment (Segment 2) has a length of $1.0\text{ m}$ and an area of $400\text{ mm}^2$. The bar is suspended from the ceiling. It carries a downward point load of $30\text{ kN}$ at the junction between the segments, and a downward point load of $20\text{ kN}$ at the free bottom end. Find the total elongation ($E=70\text{ GPa}$).

Problem: A vertical heavy steel cable of length $50\text{ m}$ and cross-sectional area $100\text{ mm}^2$ hangs freely from a tower. The unit weight of steel is $\gamma=78.5\text{ kN/m}^3$. Calculate the elongation strictly due to its own weight. Assume $E=200\text{ GPa}$.

Problem: A continuous brass pipeline is exactly $3\text{ m}$ long at an ambient temperature of $20^\circ\text{C}$. Find its new length if hot fluid causes the temperature to rise to $80^\circ\text{C}$. The coefficient of linear thermal expansion for brass is $\alpha=19 \times 10^{-6} /^\circ\text{C}$.

Problem: A steel railroad rail is $10\text{ m}$ long and is laid securely between two massive concrete abutments at $15^\circ\text{C}$ with absolutely zero gap. Determine the internal compressive stress in the rails if the summer temperature rises to $50^\circ\text{C}$ and the extreme pressure causes the concrete abutments to yield (get pushed outward) by exactly $1.0\text{ mm}$. ($E=200\text{ GPa}$, $\alpha=11.7 \times 10^{-6} /^\circ\text{C}$)

Problem: A solid circular steel drive shaft has a diameter of $40\text{ mm}$ and a length of $1.5\text{ m}$. It is subjected to a twisting moment (torque) of $500\text{ N}\cdot\text{m}$. Determine the angle of twist experienced by the shaft in degrees. Assume the Shear Modulus is $G=80\text{ GPa}$.