A beam is supported by a pin at one end and a roller at the other. Before writing any equilibrium equations, how many unknown reaction components does this system have, and is it statically determinate?
A2 unknowns (one from each support) — statically indeterminate because there aren't enough equations
B4 unknowns (two from each support) — statically indeterminate, requiring additional equations
C3 unknowns (two from the pin, one from the roller) — statically determinate
D3 unknowns (two from the pin, one from the roller) — statically indeterminate
A pin provides two reaction components (Rx and Ry); a roller provides one (normal to its surface). Total: 3 unknowns. In 2D, there are exactly 3 equilibrium equations (ΣFx = 0, ΣFy = 0, ΣM = 0), so the system is statically determinate — solvable with equilibrium alone. Option B is the classic error of assigning a moment reaction to the pin, giving 3 + 1 = 4 unknowns. Pins allow free rotation and provide no moment reaction.
Question 2 Multiple Choice
A wall-mounted cantilever beam has a fixed (cantilever) support at the wall and a roller support under the free end. How many total unknown reaction components must be solved for?
A3 — one from the roller and two from the fixed support
B4 — two force components and one moment from the fixed support, plus one force from the roller
C2 — only the vertical forces matter in beam problems
D3 — the moment at the fixed support cancels the roller reaction, leaving three independent unknowns
A fixed support provides three unknowns: two force components (Rx, Ry) and a moment reaction (M). A roller provides one unknown force. Total: 4 unknowns. With only 3 equilibrium equations available, this is statically indeterminate — you cannot solve it with equilibrium alone and need compatibility equations from mechanics of materials. Option A is the most common error: omitting the moment reaction at the fixed support, which is the defining feature that distinguishes it from a pin.
Question 3 True / False
A pin support can resist forces in both the x and y directions, as well as rotational moments about its contact point.
TTrue
FFalse
Answer: False
A pin support constrains translation in x and y (two force reactions) but allows free rotation — it provides NO moment reaction. This is the definition of a pin (hinge): it permits rotation at the connection point. Only a fixed support resists all three: Rx, Ry, and moment M. Incorrectly assigning a moment reaction to a pin gives 4 unknowns instead of 3 for a simply supported beam, making a determinate problem appear indeterminate.
Question 4 True / False
A roller support on a horizontal surface provides only a vertical reaction force, with no horizontal force component and no moment reaction.
TTrue
FFalse
Answer: True
A roller constrains motion perpendicular to its rolling surface and allows free motion parallel to that surface. On a horizontal surface, the roller prevents vertical displacement (one unknown: the normal force, vertical), allows horizontal sliding, and allows rotation. There is no horizontal reaction and no moment reaction. This is why a simply supported beam (pin + roller) has exactly 3 unknowns total. If the roller surface were inclined, the single reaction force would be normal to that inclined surface.
Question 5 Short Answer
Why should you count unknown reaction components before writing equilibrium equations, and what does the count tell you about your solution approach?
Think about your answer, then reveal below.
Model answer: Counting unknowns tells you whether the problem is statically determinate before you commit to a solution approach. In 2D, you have exactly 3 equilibrium equations. If you have exactly 3 unknowns, the system is determinate and solvable with equilibrium alone. If you have more than 3 unknowns, the system is statically indeterminate and requires additional equations from material behavior (compatibility conditions). Discovering indeterminacy after you've written and tried to solve the equations wastes time and causes confusion — the count is a prerequisite check.
The count also guides how you set up the equations strategically. With 3 unknowns, you can eliminate two at once by taking moments about a point where two unknowns act — leaving one equation with one unknown. For a pin-roller beam, taking moments about the pin location eliminates both pin force components, letting you solve directly for the roller reaction from a single equation. This strategic use of moment equations is only possible when you know the count and locations of unknowns before you start writing. Identifying support type → counting unknowns → checking determinacy → writing equations is the correct sequence.