Scenario: A steel truss bridge carries a significant load from traffic. The engineers must ensure that the internal forces developed within the individual truss members do not exceed the material's capacity.

Analysis: When a heavy truck drives over the bridge, the external load creates a complex system of internal forces throughout the truss structure. Some members will stretch, indicating they are in tension, and other members will compress, meaning they are in compression. The internal force distributed continuously throughout the material's cross-section is what we define as stress. The structure maintains equilibrium by distributing these internal stresses through its nodes.

Scenario: A tower crane lifts a heavy concrete block at a construction site.

Analysis: As the crane lifts the block, the steel cable suspending it experiences an immense internal tensile force trying to pull it apart. Simultaneously, the vertical mast of the crane experiences internal compressive forces pushing down on it, as well as bending forces. Understanding these internal forces is critical because if the internal force (stress) exceeds the material's failure limit, the cable could snap or the mast could buckle, leading to a catastrophic failure.

A steel rod with a circular cross-section of radius 10 mm is subjected to a tensile force of 50 kN. Determine the normal stress in the rod.

A hollow bronze cylinder with an outer diameter of 50 mm and an inner diameter of 30 mm carries an axial compressive load of 120 kN. Calculate the normal stress in the cylinder.

An aluminum tube is rigidly attached between a bronze rod and a steel rod. Axial loads are applied at the positions indicated. Determine the maximum normal stress in the assembly if the cross-sectional areas are: Bronze ($A_b = 400\text{ mm}^2$), Aluminum ($A_a = 600\text{ mm}^2$), and Steel ($A_s = 300\text{ mm}^2$). The loads are: $P_1 = 20\text{ kN}$ pulling the bronze to the left, $P_2 = 15\text{ kN}$ pulling the connection of bronze/aluminum to the right, and $P_3 = 10\text{ kN}$ pulling the connection of aluminum/steel to the right. The steel is attached to a wall on the right.

A single bolt connects two plates and is pulled by a force $P = 30\text{ kN}$. If the bolt has a diameter of $16\text{ mm}$, determine the shear stress in the bolt.

A bolt connects three plates. The inner plate is pulled with a force of $40\text{ kN}$, while the two outer plates resist the force equally. If the bolt diameter is $12\text{ mm}$, determine the shear stress in the bolt.

A circular punch $20\text{ mm}$ in diameter is used to punch a hole through a $10\text{ mm}$ thick steel plate. If the force required to punch the hole is $250\text{ kN}$, what is the maximum shear stress in the plate?

A $20\text{ mm}$ diameter bolt connects two steel plates, each $15\text{ mm}$ thick. A tensile load of $45\text{ kN}$ is applied to the connection. Determine the bearing stress between the bolt and the plate.

A square column $300\text{ mm}$ on each side rests on a concrete footing $1.2\text{ m} \times 1.2\text{ m}$. If the load on the column is $800\text{ kN}$, calculate the bearing stress between the column and the footing, and the bearing stress on the soil beneath the footing.

A lap joint uses four $16\text{ mm}$ bolts to connect two $12\text{ mm}$ thick plates. The joint transmits a load of $150\text{ kN}$. Calculate the average bearing stress on the bolts.

A structural steel rod has an ultimate tensile strength of $400\text{ MPa}$. If the required Factor of Safety is $2.5$, determine the allowable working stress.

A solid circular rod must carry a tensile load of $60\text{ kN}$. The material has a yield strength of $250\text{ MPa}$. Using a Factor of Safety of $2.0$ against yielding, calculate the required diameter of the rod.

Two plates are to be joined by a single bolt. The applied load is $50\text{ kN}$. The allowable shear stress for the bolt material is $100\text{ MPa}$ and the allowable bearing stress for the plate material is $150\text{ MPa}$. The plates are $10\text{ mm}$ thick. What is the minimum required bolt diameter to satisfy both shear and bearing constraints?