Questions: Support Reactions and Classifications

Question 1

A simply-supported beam rests on a pin at its left end and a roller at its right end. How many unknown reaction components must be solved for in total?

• A. Two — one vertical force at each support
• B. Three — two at the pin and one at the roller
• C. Four — two at each support
• D. Five — two at the pin, one at the roller, and two from the beam's self-weight

Answer: (B)

A pin prevents translation in both directions and introduces two unknown force components (horizontal and vertical). A roller prevents only perpendicular translation and introduces one unknown (the normal reaction). Total: 3 unknowns. This matches exactly the three equilibrium equations available in 2D (ΣFx = 0, ΣFy = 0, ΣM = 0), making this structure statically determinate. Note that the beam's self-weight is a known applied load, not an unknown reaction.

Question 2

A cantilever beam is embedded in a wall at one end (fixed support) with the other end free. Compared to a pin support, what additional reaction does the fixed support introduce?

• A. An additional vertical force component to resist bending
• B. A reaction moment that resists rotation of the beam at the wall
• C. A horizontal tension force along the beam's axis
• D. Two additional force components, bringing the total to four unknowns

Answer: (B)

A pin prevents translation (2 force components) but allows free rotation — it cannot resist a moment. A fixed support prevents both translation and rotation, adding a reaction moment (couple) to the two force components. This moment physically represents the wall's resistance to the beam rotating at its root. The fixed support therefore has 3 unknowns (Rx, Ry, M), compared to 2 for a pin. This reaction moment is what makes cantilever structures possible: the wall must absorb not just force but also the rotational tendency from the applied loads.

Question 3

A pin support prevents the connected beam from rotating at the support point, just as a fixed (cantilever) support does.

• T. True
• F. False

Answer: False

This is the most important distinction between pins and fixed supports. A pin allows free rotation — the beam can rotate about the pin. It only prevents translation (movement in any direction). A fixed support prevents both translation and rotation, which is why it generates a reaction moment in addition to the two force components. Getting this wrong leads to incorrectly modeling structures: a pin-roller beam can deflect and rotate freely at both ends, whereas a fixed support locks the beam's end slope.

Question 4

A roller placed on an inclined surface at 30° to horizontal provides a reaction force that is perpendicular to that inclined surface, not necessarily vertical.

• T. True
• F. False

Answer: True

A roller can slide freely along the surface it sits on but cannot pass through it. The reaction force is always perpendicular (normal) to the contact surface, opposing the tendency to penetrate it. If the surface is inclined at 30°, the reaction is perpendicular to that incline — at 60° from horizontal, not vertical. This is why the orientation of the roller's surface matters for free-body diagrams: the direction of the reaction force changes with the surface angle.

Question 5

Explain how counting support unknowns before writing equilibrium equations tells you whether a structure is statically determinate, and why this step matters.

Model answer: Each support type introduces a fixed number of unknown reactions: roller = 1, pin = 2, fixed = 3 (in 2D). Sum these across all supports to get the total unknowns. In 2D statics, there are exactly 3 equilibrium equations (ΣFx = 0, ΣFy = 0, ΣM = 0). If total unknowns = 3, the structure is statically determinate — solvable by equilibrium alone. If unknowns > 3, it is statically indeterminate and requires additional compatibility equations. If unknowns < 3, it is a mechanism. This check costs nothing and immediately reveals whether the problem is solvable before any equations are written.

Students often jump to writing equations and get stuck, not realizing the structure was indeterminate from the start. The degree of static indeterminacy = (unknowns − 3), which tells you how many additional conditions (deformation compatibility, material properties) are needed. For a simply-supported beam: pin (2) + roller (1) = 3 = 3 equations → determinate. For a beam fixed at both ends: fixed (3) + fixed (3) = 6 unknowns, degree 3 indeterminate.