A horizontal beam of length $10.0\text{ m}$ 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 $4.00\text{ m}$ from point A. Neglecting the weight of the beam, determine the reactions at supports A and B.

A simply supported beam spans $6.00\text{ m}$ between a pin support at A (left) and a roller support at B (right). It carries a uniform distributed load (UDL) of $2.00\text{ kN/m}$ 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 $0.500\text{ m}$ above A. Point C is $1.00\text{ m}$ to the right of A. The short link at C acts at a $30.0^\circ$ angle above the horizontal. Find the reaction at A.

A cantilever beam of length $5.00\text{ m}$ is fixed to a wall at point A. A downward point load of $15.0\text{ kN}$ is applied at the free end (point B), and a clockwise concentrated moment of $20.0\text{ kN}\cdot\text{m}$ is applied at the midpoint ($2.50\text{ m}$ from A). Determine the reaction forces and moment at the fixed support A.

A beam of length $8.00\text{ m}$ is supported by a pin at A (left end) and a roller at B (right end). It carries a triangular distributed load that increases from $0\text{ kN/m}$ at A to $6.00\text{ kN/m}$ at B. Find the reactions at supports A and B.

An overhanging beam spans $10.0\text{ m}$ total. It is supported by a pin at A ($x = 2.00\text{ m}$) and a roller at B ($x = 8.00\text{ m}$). A point load of $10.0\text{ kN}$ is applied at the left end ($x = 0\text{ m}$), and a point load of $15.0\text{ kN}$ is applied at the right end ($x = 10.0\text{ m}$). A uniform load of $3.00\text{ kN/m}$ is applied between the supports (from $x = 2.00\text{ m}$ to $x = 8.00\text{ m}$). Find the reactions.

A frame consists of a vertical member AC pinned to the ground at A. A horizontal member BC is rigidly attached at C. The member BC has length $4.00\text{ m}$. At point B, a frictionless pulley is attached. A cable passes over the pulley and supports a mass of $100.\text{ kg}$. The other end of the cable is anchored vertically to the ground. Determine the reactions at A. Assume the pulley has negligible radius and mass. $g = 9.81\text{ m/s}^2$.

A uniform ladder of length $5.00\text{ m}$ and weight $400.\text{ N}$ rests against a smooth vertical wall at point B and on a rough horizontal floor at point A. The base of the ladder is $3.00\text{ m}$ from the wall. Find the normal force exerted by the wall at B, and the normal and frictional forces exerted by the floor 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.

Balconies protruding from buildings without visible underneath columns are designed as cantilever structures. Discuss the type of reaction forces generated at the wall and why a fixed support is mathematically and physically required.

In the analysis of planar rigid bodies subjected to exactly three non-parallel forces, the lines of action of these forces must intersect at a common point if the body is in equilibrium. Discuss how this principle is applied to determine unknown force directions.