A horizontal beam of length $10\text{ m}$ is supported by a pin at end A and a roller at end B. A concentrated downward load of $500\text{ N}$ is applied at $4\text{ m}$ from point A. Neglecting the weight of the beam, determine the reactions at supports A and B.

A simply supported beam spans $6 \text{ m}$ between a pin support at A (left) and a roller support at B (right). It carries a uniform distributed load (UDL) of $2 \text{ kN/m}$ over its entire length. Determine the reactions at A and B.

A rigid L-shaped bracket is pinned at the corner A and supported by a short link at the end of the horizontal arm (point C). A horizontal force of $500 \text{ N}$ acting to the left is applied at the end of the vertical arm (point B). Point B is $0.5 \text{ m}$ above A. Point C is $1 \text{ m}$ to the right of A. The short link at C acts at a $30^\circ$ angle above the horizontal. Find the reaction at A.

Most long-span bridges, such as highway overpasses, are constructed with a pin support at one end and a roller support at the other, rather than being pinned or fixed at both ends. Discuss the engineering rationale behind this design choice from the perspective of rigid body equilibrium.

When analyzing a truss structure, engineers make the fundamental assumption that all members are "two-force members." Explain what a two-force member is, the conditions required for a member to be considered one, and how this simplifies the equilibrium analysis of the entire truss.