Solved Problems

A steel rod with a cross-sectional area of $200 \text{ mm}^2$ and a length of $5 \text{ m}$ is subjected to an axial tensile load of $50 \text{ kN}$. Determine the elongation of the rod. Assume $E = 200 \text{ GPa}$.

Two plates are connected by a single $20 \text{ mm}$ diameter bolt in a lap joint. If a tensile force of $100 \text{ kN}$ is applied to the plates, determine the average shear stress in the bolt.

A steel bar is rigidly held between two unyielding supports at $20^\circ\text{C}$. If the temperature is raised to $60^\circ\text{C}$, determine the thermal stress developed in the bar. Assume $\alpha = 11.7 \times 10^{-6} /^\circ\text{C}$ and $E = 200 \text{ GPa}$.

A light fixture weighing $500 \text{ kg}$ is supported by a vertical steel rod. If the allowable tensile stress is $100 \text{ MPa}$, determine the minimum required diameter of the rod. (Use $g = 9.81 \text{ m/s}^2$)

A heavy uniform vertical steel rod of length $L = 10 \text{ m}$, cross-sectional area $A = 500 \text{ mm}^2$, and density $\rho = 7850 \text{ kg/m}^3$ hangs freely from the ceiling. Determine the total elongation due to its own weight. Assume $E = 200 \text{ GPa}$ and $g = 9.81 \text{ m/s}^2$.

A tie rod is intended to support a tensile load of $40 \text{ kN}$. It is made of a steel alloy with a yield strength of $250 \text{ MPa}$ and an ultimate strength of $400 \text{ MPa}$. If a factor of safety of $2.0$ with respect to the yield strength is required, determine the minimum diameter of the rod. What is the Margin of Safety with respect to ultimate failure?

A steel bar of length $L = 2 \text{ m}$ is placed between two rigid walls. A point load $P = 100 \text{ kN}$ is applied at a distance $a = 0.5 \text{ m}$ from the left wall. The bar has a uniform cross-sectional area $A = 500 \text{ mm}^2$ and modulus of elasticity $E = 200 \text{ GPa}$. Determine the support reactions at the left and right walls.

A rectangular rubber block has a width of $100 \text{ mm}$, a height of $50 \text{ mm}$, and a depth of $200 \text{ mm}$. A uniform shear force $V = 5 \text{ kN}$ is applied to the top surface while the bottom surface is rigidly fixed. The top surface is observed to undergo a horizontal displacement of $2.5 \text{ mm}$. Determine the shear strain and the shear modulus of the rubber.

A steel column rests on a concrete foundation pad. The column carries an axial compressive load of $400 \text{ kN}$. If the allowable bearing stress for the concrete is $8.00 \text{ MPa}$, determine the minimum required square area and the side length of the base plate needed to distribute the load safely.

Case Study: An aerospace engineering team is selecting a material for a landing gear strut. The strut must withstand high impact loads without experiencing permanent deformation, and it must also be lightweight to maximize fuel efficiency. They are considering steel, titanium, and an aluminum alloy.

Case Study: During the early days of railway construction, sections of steel track were laid end-to-end with no gaps. During a particularly hot summer, multiple sections of the track buckled out of shape, causing train derailments. Why did this happen, and how is it prevented in modern engineering?

Case Study: A cylindrical pressure vessel used in a chemical plant ruptured catastrophically. Post-accident investigation revealed that the failure originated at a sharp, right-angled corner where a nozzle pipe intersected the main cylindrical body, even though the overall calculated wall stress was well below the material's yield strength.

Case Study: During a standard tensile test, a specimen of low-carbon steel is stretched until it breaks. Engineers observe that before the final rupture, the force required to stretch the material actually decreases, and a localized narrowing of the specimen occurs.